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

Full Article  |  PDF Article
OSA Recommended Articles
Stokes–anti-Stokes correlations in diamond

Mark Kasperczyk, Ado Jorio, Elke Neu, Patrick Maletinsky, and Lukas Novotny
Opt. Lett. 40(10) 2393-2396 (2015)

Control of quantum transverse correlations on a four-photon system

P.-L. de Assis, M. A. D. Carvalho, L. P. Berruezo, J. Ferraz, I. F. Santos, F. Sciarrino, and S. Pádua
Opt. Express 19(4) 3715-3729 (2011)

Photon statistics and polarization correlations at telecommunications wavelengths from a warm atomic ensemble

R. T. Willis, F. E. Becerra, L. A. Orozco, and S. L. Rolston
Opt. Express 19(15) 14632-14641 (2011)

More Recommended Articles

References

  • View by:
  • Article Order
  • |
  • Year
  • |
  • Author
  • |
  • Publication
  1. 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]
  2. 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]
  3. 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]
  4. C. V. Bennett and B. H. Kolner, “Upconversion time microscope demonstrating 103 × magnification of femtosecond waveforms,” Opt. Lett. 24, 783–785 (1999).
    [Crossref]
  5. K. A. O’Donnell and A. B. U’Ren, “Time-resolved up-conversion of entangled photon pairs,” Phys. Rev. Lett. 103, 123602 (2009).
    [Crossref]
  6. J. Gea-Banacloche, “Two-photon absorption of nonclassical light,” Phys. Rev. Lett. 62, 1603–1606 (1989).
    [Crossref] [PubMed]
  7. J. Javanainen and P. L. Gould, “Linear intensity dependence of a two-photon transition rate,” Phys. Rev. A. 41, 5088–5091 (1990).
    [Crossref] [PubMed]
  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. 75, 3426–3429 (1995).
    [Crossref] [PubMed]
  9. 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).
  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. 30, 314–316 (2005)
    [Crossref] [PubMed]
  11. 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]
  12. R. Shimizu and K. Edamatsu, “High-flux and broadband biphoton sources with controlled frequency entanglement,” Opt. Express 17, 16385–16393 (2009).
    [Crossref] [PubMed]
  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. 68043813 (2003).
    [Crossref]
  14. M. Bixon and J. Jortner, “Long radiative lifetimes of small molecules,” J. Chem. Phys. 50, 3284–3290 (1969).
    [Crossref]
  15. C. W. Gardiner, Quantum Noise (Springer-Verlag, Berlin, 1991).
  16. See, for example, D. N. Klyshko, Photons and Nonlinear Optics (Gordon and Breach Science, New York, 1988).
  17. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 592044–2046 (1987).
    [Crossref] [PubMed]
  18. 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.
  19. H. Oka, “Efficient selective two-photon excitation by tailored quantum-correlated photons,” Phys. Rev. A 81, 063819 (2010).
    [Crossref]
  20. 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]
  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. 89, 224–228 (2009).
    [Crossref]

2010 (2)

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]

2009 (3)

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]

R. Shimizu and K. Edamatsu, “High-flux and broadband biphoton sources with controlled frequency entanglement,” Opt. Express 17, 16385–16393 (2009).
[Crossref] [PubMed]

K. A. O’Donnell and A. B. U’Ren, “Time-resolved up-conversion of entangled photon pairs,” Phys. Rev. Lett. 103, 123602 (2009).
[Crossref]

2007 (1)

2005 (2)

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]

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]

2003 (1)

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. 68043813 (2003).
[Crossref]

2002 (1)

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]

2000 (1)

1999 (1)

1995 (1)

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]

1990 (1)

J. Javanainen and P. L. Gould, “Linear intensity dependence of a two-photon transition rate,” Phys. Rev. A. 41, 5088–5091 (1990).
[Crossref] [PubMed]

1989 (1)

J. Gea-Banacloche, “Two-photon absorption of nonclassical light,” Phys. Rev. Lett. 62, 1603–1606 (1989).
[Crossref] [PubMed]

1987 (1)

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 592044–2046 (1987).
[Crossref] [PubMed]

1969 (1)

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

1836 (1)

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).

Bennett, C. V.

Bettinelli, M.

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]

Biswas, A.

Bixon, M.

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

Boyer, J

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]

Capobianco, J. A.

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]

Carrasco, S.

Edamatsu, K.

R. Shimizu and K. Edamatsu, “High-flux and broadband biphoton sources with controlled frequency entanglement,” Opt. Express 17, 16385–16393 (2009).
[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]

Friend, C. S.

Gardiner, C. W.

C. W. Gardiner, Quantum Noise (Springer-Verlag, Berlin, 1991).

Gea-Banacloche, J.

J. Gea-Banacloche, “Two-photon absorption of nonclassical light,” Phys. Rev. Lett. 62, 1603–1606 (1989).
[Crossref] [PubMed]

Georgiades, N. P.

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]

Giovannetti, V.

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).

Gould, P. L.

J. Javanainen and P. L. Gould, “Linear intensity dependence of a two-photon transition rate,” Phys. Rev. A. 41, 5088–5091 (1990).
[Crossref] [PubMed]

Güdel, H. U.

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]

Hendrych, M.

Hofmann, H. F.

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. 68043813 (2003).
[Crossref]

Hong, C. K.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 592044–2046 (1987).
[Crossref] [PubMed]

Javanainen, J.

J. Javanainen and P. L. Gould, “Linear intensity dependence of a two-photon transition rate,” Phys. Rev. A. 41, 5088–5091 (1990).
[Crossref] [PubMed]

Jortner, J.

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

Kalashnikov, D. A.

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]

Kapoor, R.

Katamadze, K. G.

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]

Kimble, H. J.

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]

Klyshko, D. N.

See, for example, D. N. Klyshko, Photons and Nonlinear Optics (Gordon and Breach Science, New York, 1988).

Kojima, K.

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. 68043813 (2003).
[Crossref]

Kolner, B. H.

Kulik, S. P.

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]

Maccone, L.

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).

Macià, F.

Mandel, L.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 592044–2046 (1987).
[Crossref] [PubMed]

Micuda, M.

O’Donnell, K. A.

K. A. O’Donnell and A. B. U’Ren, “Time-resolved up-conversion of entangled photon pairs,” Phys. Rev. Lett. 103, 123602 (2009).
[Crossref]

Oka, H.

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]

Ou, Z. Y.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 592044–2046 (1987).
[Crossref] [PubMed]

Parkins, A. S.

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]

Polzik, E. S.

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]

Prasad, P. N.

Richards, B. S.

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]

Sasaki, K.

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. 68043813 (2003).
[Crossref]

Shalav, A.

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]

Shapiro, J. H.

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).

Shimizu, R.

Speghini, A.

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]

Takeuchi, S.

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. 68043813 (2003).
[Crossref]

Torner, L.

Torres, J. P.

Trupke, T.

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]

U’Ren, A. B.

K. A. O’Donnell and A. B. U’Ren, “Time-resolved up-conversion of entangled photon pairs,” Phys. Rev. Lett. 103, 123602 (2009).
[Crossref]

Vetrone, F.

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]

Wong, F. N. C.

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).

Würfel, P.

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]

Appl. Phys. Lett. (2)

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]

J. Chem. Phys. (1)

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

JETP Lett. (1)

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]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. A (2)

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]

Phys. Rev. A. (2)

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. 68043813 (2003).
[Crossref]

J. Javanainen and P. L. Gould, “Linear intensity dependence of a two-photon transition rate,” Phys. Rev. A. 41, 5088–5091 (1990).
[Crossref] [PubMed]

Phys. Rev. Lett. (5)

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).

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]

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 592044–2046 (1987).
[Crossref] [PubMed]

Other (3)

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.

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).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

(a) Schematic of the sequential two-step up-conversion geometry. (b) One-dimensional atom model.

Download Full Size | PPT Slide | PDF

Fig. 2
Fig. 2

|ψ2p|2 for (a) uncorrelated photons, (b) TB photons, and (c) DB photons. (d) Spectra corresponding to (a), (b), and (c). Δ = 0 and σ ≈ 30λ, where λ = 2π/k0.

Download Full Size | PPT Slide | PDF

Fig. 3
Fig. 3

〈mA〉 and 〈e〉 as a function of r/σ for (a) uncorrelated photons, (b) TB photons, and (c) DB photons.

Download Full Size | PPT Slide | PDF

Fig. 4
Fig. 4

(a) ξ as a function of Δ/σ. (b) ξ as a function of σ/λ for Δ/σ = 0 and 10.

Download Full Size | PPT Slide | PDF

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

H ^ = ∫ d k   k a ^ k † a ^ k + ω e | e 〉 〈 e | + ∑ ℓ = A , B ω m ℓ | m ℓ 〉 〈 m ℓ | + ∫ d k Γ mg π ( | m A 〉 〈 g | a ^ k + H . c . ) + ∫ d k Γ em π ( | e 〉 〈 m B | a ^ k + H . c . ) + ∫ d k Γ eg π ( | e 〉 〈 g | a ^ k + H . c . ) + H ^ relax .
H ^ relax = ∫ dq   q p ^ q † p ^ q + ∫ dq γ π ( | m A 〉 〈 m B | p ^ q + H . c . ) ,
| Ψ ( t ) 〉 = exp ( − i H ^ t ) | Ψ ( 0 ) 〉 ,
| Ψ ( 0 ) 〉 = 1 2 ∫ dk ∫ d k ′ ψ 2 p ( k , k ′ ) a ^ k † a ^ k ′ † | 0 〉 | g 〉 ,
ψ 2 p ( k , k ′ ) = ψ ( k ) ψ ( k ′ ) e − ik r 0 e − i k ′ r ′ 0 ,
ψ 2 p ( k , k ′ ) = ψ ( k ) δ ( k + k ′ − 2 k 0 ) e − i k r 0 e − i k ′ r ′ 0 ,
ψ 2 p ( k , k ′ ) = ψ ( k ) δ ( k − k ′ ) e − ik r 0 e − i k ′ r ′ 0 ,
ψ ( k ) e − ik r 0 = 1 2 π ∫ − ∞ ∞ d r ψ ( r ) exp ( − ikr )
ψ ( r ) ∝ exp [ − ( r − r 0 ) 2 / σ 2 + i k 0 ( r − r 0 ) ] ,

Metrics