## Efficient two-step up-conversion by quantum-correlated photon pairs |

Optics Express, Vol. 18, Issue 25, pp. 25839-25846 (2010)

http://dx.doi.org/10.1364/OE.18.025839

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### Abstract

We theoretically investigate the sequential two-step up-conversion of correlated photon pairs with positive and negative energy correlations, in terms of how the up-conversion efficiency depends on the incident pulse delay. A three-level atomic system having a metastable state is used to evaluate the up-conversion efficiency. It is shown that a photon pair with a positive energy correlation can drastically enhance the up-conversion efficiency compared with uncorrelated photons and correlated photons with a negative energy correlation.

© 2010 Optical Society of America

## 1. Introduction

1. R. Kapoor, C. S. Friend, A. Biswas, and P. N. Prasad, “Highly efficient infrared-to-visible energy upconversion in Er^{3+}:Y_{2}O_{3},” Opt. Lett. **25**, 338–340 (2000). [CrossRef]

2. F. Vetrone, J Boyer, J. A. Capobianco, A. Speghini, and M. Bettinelli, “980 nm excited upconversion in an Er-doped ZnO-TeO_{2} glass,” Appl. Phys. Lett. **80**, 1752–1754 (2002). [CrossRef]

3. A. Shalav, B. S. Richards, T. Trupke, P. Würfel, and H. U. Güdel, “Application of NaYF_{4}:Er^{3+} up-converting phosphors for enhanced near-infrared silicon solar cell response,” Appl. Phys. Lett. **86**, 013505 (2005). [CrossRef]

4. C. V. Bennett and B. H. Kolner, “Upconversion time microscope demonstrating 103 × magnification of femtosecond waveforms,” Opt. Lett. **24**, 783–785 (1999). [CrossRef]

## 2. Model

### 2.1. One-dimensional atom model

*r*-axis and penetrate into a one-sided microcavity (at dumping rate

*κ*). The two photons interact with an atom inside the cavity through a cavity field (at coupling rate

*g*) and an up-converted photon returns to the initial photon field. The atomic system inside the cavity is a modified three-level system including a metastable state: the ground state |

*g*〉, the two intermediate states |

*m*

_{A}〉 and |

*m*

_{B}〉, and the excited state |

*e*〉. |

*m*

_{A}〉 absorbs photons and its population rapidly decays to |

*m*

_{B}〉 through multiphonon non-radiative relaxation. |

*m*

_{B}〉 is metastable and becomes a step ladder in the transition to |

*e*〉. The energies of the atomic system are denoted by

*ω*

_{mA(B)}and

*ω*(in units of

_{e}*h̄*).

*ω*

_{mA}and

*ω*–

_{e}*ω*

_{mB}are set to the central energy

*k*

_{0}of incident photons so that |

*m*

_{A}〉 and |

*e*〉 can be resonantly excited.

*κ*≫

*g*, photons inside the cavity are emitted so rapidly from the cavity that the cavity field can be adiabatically eliminated. In this case, the atom-photon interaction can be characterized by a single effective emission rate, Γ ≡

*g*

^{2}/

*κ*. If Γ is much larger than the spontaneous emission rate of the atomic system into free space (non-cavity modes), the system can be reduced to a one-dimensional input-output system with negligible radiative losses [Fig. 1(b)], called the one-dimensional atom model.

### 2.2. Hamiltonian and quantum dynamics

*h̄*=

*c*= 1, the Hamiltonian of the whole system is given by

*k*,

*p̂*is the annihilation operator of a phonon with energy

_{q}*q*, and

*γ*is the non-radiative decay rate. Eq. (2) is a fully quantum-mechanical description of the Bixon-Jortner theory [14

14. M. Bixon and J. Jortner, “Long radiative lifetimes of small molecules,” J. Chem. Phys. **50**, 3284–3290 (1969). [CrossRef]

*g*〉 and |

*m*

_{B}〉 and between |

*m*

_{A}〉 and |

*e*〉, for simplicity.

*ψ*

_{2p}is the two-photon joint amplitude of the incident pulse. The whole wave function is normalized to be 〈Ψ|Ψ〉 = 1. The populations of the excited and intermediate atomic states are described by 〈

*e*〉 = Tr[|

*e*〉〈

*e*||Ψ(

*t*)〉〈Ψ(

*t*)|] and 〈

*m*〉 = Tr[|

_{ℓ}*m*〉〈

_{ℓ}*m*||Ψ(

_{ℓ}*t*)〉〈Ψ(

*t*)|], respectively.

### 2.3. Quantum-correlated photons

13. H. F. Hofmann, K. Kojima, S. Takeuchi, and K. Sasaki, “Entanglement and four-wave mixing effects in the dissipation-free nonlinear interaction of two photons at a single atom,” Phys. Rev. A. **68**043813 (2003). [CrossRef]

*ψ*(

*k*) by the Fourier transformation of the space domain and choose

*ψ*(

*k*) having a Gaussian shape, with where

*σ*is the coherent length of the wave packet. The two-photon joint spectra |

*ψ*

_{2p}|

^{2}corresponding to Eqs. (5), (6), and (7) are shown in Figs. 2(a), 2(b), and 2(c), respectively. Intriguingly, the spectra of these photon pairs are identical in terms of classical electromagnetic field, as shown in Fig. 2(d). Thus, the only difference is the quantum correlation, which can be controlled only by quantizing light fields.

## 3. Results

19. H. Oka, “Efficient selective two-photon excitation by tailored quantum-correlated photons,” Phys. Rev. A **81**, 063819 (2010). [CrossRef]

*e*〉 for smaller Δ, especially for Δ = 0.

## 4. Summary and discussion

## Acknowledgments

## References and links

1. | R. Kapoor, C. S. Friend, A. Biswas, and P. N. Prasad, “Highly efficient infrared-to-visible energy upconversion in Er |

2. | F. Vetrone, J Boyer, J. A. Capobianco, A. Speghini, and M. Bettinelli, “980 nm excited upconversion in an Er-doped ZnO-TeO |

3. | A. Shalav, B. S. Richards, T. Trupke, P. Würfel, and H. U. Güdel, “Application of NaYF |

4. | C. V. Bennett and B. H. Kolner, “Upconversion time microscope demonstrating 103 × magnification of femtosecond waveforms,” Opt. Lett. |

5. | K. A. O’Donnell and A. B. U’Ren, “Time-resolved up-conversion of entangled photon pairs,” Phys. Rev. Lett. |

6. | J. Gea-Banacloche, “Two-photon absorption of nonclassical light,” Phys. Rev. Lett. |

7. | J. Javanainen and P. L. Gould, “Linear intensity dependence of a two-photon transition rate,” Phys. Rev. A. |

8. | N. P. Georgiades, E. S. Polzik, K. Edamatsu, H. J. Kimble, and A. S. Parkins, “Nonclassical excitation for atoms in a squeezed vacuum,” Phys. Rev. Lett. |

9. | V. Giovannetti, L. Maccone, J. H. Shapiro, and F. N. C. Wong, “Generating entangled two-photon states with coincident frequencies,” Phys. Rev. Lett. |

10. | J. P. Torres, F. Macià, S. Carrasco, and L. Torner, “Engineering the frequency correlations of entangled two-photon states by achromatic phase matching,” Opt. Lett. |

11. | M. Hendrych, M. Micuda, and J. P. Torres, “Tunable control of the frequency correlations of entangled photons,” Opt. Lett. |

12. | R. Shimizu and K. Edamatsu, “High-flux and broadband biphoton sources with controlled frequency entanglement,” Opt. Express |

13. | H. F. Hofmann, K. Kojima, S. Takeuchi, and K. Sasaki, “Entanglement and four-wave mixing effects in the dissipation-free nonlinear interaction of two photons at a single atom,” Phys. Rev. A. |

14. | M. Bixon and J. Jortner, “Long radiative lifetimes of small molecules,” J. Chem. Phys. |

15. | C. W. Gardiner, |

16. | See, for example, D. N. Klyshko, |

17. | C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. |

18. | We adopt the energies of an Er atom, |

19. | H. Oka, “Efficient selective two-photon excitation by tailored quantum-correlated photons,” Phys. Rev. A |

20. | H. Oka, “Real-time analysis of two-photon excitation by correlated photons: Pulse-width dependence of excitation efficiency,” Phys. Rev. A |

21. | See, for example, D. A. Kalashnikov, K. G. Katamadze, and S. P. Kulik, “Controlling the spectrum of a two-photon field: Inhomogeneous broadening due to a temperature gradient,” JETP Lett. |

**OCIS Codes**

(270.0270) Quantum optics : Quantum optics

(270.4180) Quantum optics : Multiphoton processes

(270.5580) Quantum optics : Quantum electrodynamics

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: September 21, 2010

Revised Manuscript: October 4, 2010

Manuscript Accepted: October 5, 2010

Published: November 24, 2010

**Citation**

Hisaki Oka, "Efficient two-step up-conversion by quantum-correlated photon pairs," Opt. Express **18**, 25839-25846 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-25-25839

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### References

- R. Kapoor, C. S. Friend, A. Biswas, and P. N. Prasad, "Highly efficient infrared-to-visible energy upconversion in Er3+:Y2O3," Opt. Lett. 25, 338-340 (2000). [CrossRef]
- F. Vetrone, J. Boyer, J. A. Capobianco, A. Speghini, and M. Bettinelli, "980 nm excited upconversion in an Er-doped ZnO-TeO2 glass," Appl. Phys. Lett. 80, 1752-1754 (2002). [CrossRef]
- A. Shalav, B. S. Richards, T. Trupke, P. Würfel, and H. U. Güdel, "Application of NaYF4:Er3+ up-converting phosphors for enhanced near-infrared silicon solar cell response," Appl. Phys. Lett. 86, 013505 (2005). [CrossRef]
- C. V. Bennett, and B. H. Kolner, "Upconversion time microscope demonstrating 103 × magnification of femtosecond waveforms," Opt. Lett. 24, 783-785 (1999). [CrossRef]
- K. A. O’Donnell and A. B. U’Ren, "Time-resolved up-conversion of entangled photon pairs," Phys. Rev. Lett. 103, 123602 (2009). [CrossRef]
- J. Gea-Banacloche, "Two-photon absorption of nonclassical light," Phys. Rev. Lett. 62, 1603-1606 (1989). [CrossRef] [PubMed]
- J. Javanainen, and P. L. Gould, "Linear intensity dependence of a two-photon transition rate," Phys. Rev. A 41, 5088-5091 (1990). [CrossRef] [PubMed]
- N. P. Georgiades, E. S. Polzik, K. Edamatsu, H. J. Kimble, and A. S. Parkins, "Nonclassical excitation for atoms in a squeezed vacuum," Phys. Rev. Lett. 75, 3426-3429 (1995). [CrossRef] [PubMed]
- V. Giovannetti, L. Maccone, J. H. Shapiro, and F. N. C. Wong, "Generating entangled two-photon states with coincident frequencies," Phys. Rev. Lett. 88, 183602 (2002).
- J. P. Torres, F. Macià, S. Carrasco, and L. Torner, "Engineering the frequency correlations of entangled two photon states by achromatic phase matching," Opt. Lett. 30, 314-316 (2005). [CrossRef] [PubMed]
- M. Hendrych, M. Micuda, and J. P. Torres, "Tunable control of the frequency correlations of entangled photons," Opt. Lett. 32, 2339-2341 (2007). [CrossRef] [PubMed]
- R. Shimizu, and K. Edamatsu, "High-flux and broadband biphoton sources with controlled frequency entanglement," Opt. Express 17, 16385-16393 (2009). [CrossRef] [PubMed]
- H. F. Hofmann, K. Kojima, S. Takeuchi, and K. Sasaki, "Entanglement and four-wave mixing effects in the dissipation-free nonlinear interaction of two photons at a single atom," Phys. Rev. A 68, 043813 (2003). [CrossRef]
- M. Bixon, and J. Jortner, "Long radiative lifetimes of small molecules," J. Chem. Phys. 50, 3284-3290 (1969). [CrossRef]
- C. W. Gardiner, Quantum Noise (Springer-Verlag, Berlin, 1991).
- See, for example,D. N. Klyshko, Photons and Nonlinear Optics (Gordon and Breach Science, New York, 1988).
- C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of subpicosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044-2046 (1987). [CrossRef] [PubMed]
- We adopt the energies of an Er atom, 4I15/2, 4I11/2, 4I9/2, and 4F7/2 for |g〉, |mB〉, |mA〉, and |e〉, respectively. Further, we ignore influences of other energy levels for simplicity.
- H. Oka, "Efficient selective two-photon excitation by tailored quantum-correlated photons," Phys. Rev. A 81, 063819 (2010). [CrossRef]
- H. Oka, "Real-time analysis of two-photon excitation by correlated photons: Pulse-width dependence of excitation efficiency," Phys. Rev. A 81, 053837 (2010). [CrossRef]
- See, for example,D. A. Kalashnikov, K. G. Katamadze, and S. P. Kulik, "Controlling the spectrum of a two photon field: Inhomogeneous broadening due to a temperature gradient," JETP Lett. 89, 224-228 (2009). [CrossRef]

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