Abstract

In response to quantum optical fields, pairs of molecules generate coherent nonlinear spectroscopy signals. Homodyne signals are given by sums over terms each being a product of Liouville space pathways of the pair of molecules times the corresponding optical field correlation function. For classical fields all field correlation functions may be factorized and become identical products of field amplitudes. The signal is then given by the absolute square of a susceptibility which in turn is a sum over pathways of a single molecule. The molecular pathways of different molecules in the pair are uncorrelated in this case (each path of a given molecule can be accompanied by any path of the other). However, entangled photons create an entanglement between the molecular pathways. We use the superoperator nonequlibrium Green’s functions formalism to demonstrate the signatures of this pathway-entanglement in the difference frequency generation signal. Comparison is made with an analogous incoherent two-photon fluorescence signal.

© 2009 Optical Society of America

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References

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  1. Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, "Deterministic Entanglement of Two Trapped Ions," Phys. Rev. Lett. 81, 3631 (1998).
    [CrossRef]
  2. P. Walther, J. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, "De Broglie wavelength of a nonlocal four-photon state," Nature 429, 158 (2004).
    [CrossRef] [PubMed]
  3. J. Javanainen and P. Gould, "Linear intensity dependence of a two-photon transition rate," Phys. Rev. A 41, 5088 (1990).
    [CrossRef] [PubMed]
  4. B. Dayan, "Theory of two-photon interactions with broadband down-converted light and entangled photons," Phys. Rev. A 76, 43813 (2007).
    [CrossRef]
  5. D. Lee and T. Goodson, "Entangled Photon Absorption in an Organic Porphyrin Dendrimer," J. Phys. Chem. B 110, 25582 (2006).
    [CrossRef]
  6. A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, "Temporal Shaping of Entangled Photons," Phys. Rev. Lett. 94, 073601 (2005).
    [CrossRef] [PubMed]
  7. C. K. Hong and L. Mandel, "Theory of parametric frequency down conversion of light," Phys. Rev. A 31, 2409 (1985).
    [CrossRef] [PubMed]
  8. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
  9. C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University Press, 2005).
  10. H. Fei, B. Jost, S. Popescu, B. Saleh, and M. Teich, "Entanglement-Induced Two-Photon Transparency," Phys. Rev. Lett. 78, 1679-1682 (1997).
    [CrossRef]
  11. F. Lissandrin, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Quantum theory of entangled-photon photoemission," Phys. Rev. B 69, 165317 (2004).
    [CrossRef]
  12. M. Teich and B. Saleh, "Entangled-photon microscopy, spectroscopy, and display," US Patent 5,796,477 (1998).
  13. B. Saleh, B. Jost, H. Fei, and M. Teich, "Entangled-Photon Virtual-State Spectroscopy," Phys. Rev. Lett. 80, 3483 (1998).
    [CrossRef]
  14. O. Roslyak, C. Marx, and S. Mukamel, "A unified description of sum frequency generation, parametric down conversion and two photon fluorescence." J. Mol. Phys. (submitted) (2008).
  15. C. Marx, U. Harbola, and S. Mukamel, "Nonlinear optical spectroscopy of single, few, and many molecules: Nonequilibrium Green’s function QED approach," Phys. Rev. A 77, 22110 (2008).
    [CrossRef] [PubMed]
  16. R. Glauber, "The photon theory of optical coherence," Phys. Rev. 130, 2529 (1963).
    [CrossRef]
  17. R. Glauber, Quantum Theory of Optical Coherence: Selected Papers and Lectures (Wiley-VCH, 2007).
  18. O. Roslyak, C. Marx, and S. Mukamel, "Manipulating quantum pathways of matter with entangled photons," Phys. Rev. B (submitted) (2008).
  19. E. M. Nagasako, S. J. Bentley, R. W. Boyd, and G. S. Agarwal, "Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers," Phys. Rev. A 64, 043802 (2001).
    [CrossRef]
  20. S. Mukamel, Principles of nonlinear optical spectroscopy (Oxford University Press New York, 1995).

2008

O. Roslyak, C. Marx, and S. Mukamel, "A unified description of sum frequency generation, parametric down conversion and two photon fluorescence." J. Mol. Phys. (submitted) (2008).

C. Marx, U. Harbola, and S. Mukamel, "Nonlinear optical spectroscopy of single, few, and many molecules: Nonequilibrium Green’s function QED approach," Phys. Rev. A 77, 22110 (2008).
[CrossRef] [PubMed]

O. Roslyak, C. Marx, and S. Mukamel, "Manipulating quantum pathways of matter with entangled photons," Phys. Rev. B (submitted) (2008).

2007

B. Dayan, "Theory of two-photon interactions with broadband down-converted light and entangled photons," Phys. Rev. A 76, 43813 (2007).
[CrossRef]

2006

D. Lee and T. Goodson, "Entangled Photon Absorption in an Organic Porphyrin Dendrimer," J. Phys. Chem. B 110, 25582 (2006).
[CrossRef]

2005

A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, "Temporal Shaping of Entangled Photons," Phys. Rev. Lett. 94, 073601 (2005).
[CrossRef] [PubMed]

2004

P. Walther, J. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, "De Broglie wavelength of a nonlocal four-photon state," Nature 429, 158 (2004).
[CrossRef] [PubMed]

F. Lissandrin, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Quantum theory of entangled-photon photoemission," Phys. Rev. B 69, 165317 (2004).
[CrossRef]

2001

E. M. Nagasako, S. J. Bentley, R. W. Boyd, and G. S. Agarwal, "Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers," Phys. Rev. A 64, 043802 (2001).
[CrossRef]

1998

B. Saleh, B. Jost, H. Fei, and M. Teich, "Entangled-Photon Virtual-State Spectroscopy," Phys. Rev. Lett. 80, 3483 (1998).
[CrossRef]

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, "Deterministic Entanglement of Two Trapped Ions," Phys. Rev. Lett. 81, 3631 (1998).
[CrossRef]

1997

H. Fei, B. Jost, S. Popescu, B. Saleh, and M. Teich, "Entanglement-Induced Two-Photon Transparency," Phys. Rev. Lett. 78, 1679-1682 (1997).
[CrossRef]

1990

J. Javanainen and P. Gould, "Linear intensity dependence of a two-photon transition rate," Phys. Rev. A 41, 5088 (1990).
[CrossRef] [PubMed]

1985

C. K. Hong and L. Mandel, "Theory of parametric frequency down conversion of light," Phys. Rev. A 31, 2409 (1985).
[CrossRef] [PubMed]

1963

R. Glauber, "The photon theory of optical coherence," Phys. Rev. 130, 2529 (1963).
[CrossRef]

Agarwal, G. S.

E. M. Nagasako, S. J. Bentley, R. W. Boyd, and G. S. Agarwal, "Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers," Phys. Rev. A 64, 043802 (2001).
[CrossRef]

Aspelmeyer, M.

P. Walther, J. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, "De Broglie wavelength of a nonlocal four-photon state," Nature 429, 158 (2004).
[CrossRef] [PubMed]

Bentley, S. J.

E. M. Nagasako, S. J. Bentley, R. W. Boyd, and G. S. Agarwal, "Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers," Phys. Rev. A 64, 043802 (2001).
[CrossRef]

Boyd, R. W.

E. M. Nagasako, S. J. Bentley, R. W. Boyd, and G. S. Agarwal, "Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers," Phys. Rev. A 64, 043802 (2001).
[CrossRef]

Dayan, B.

B. Dayan, "Theory of two-photon interactions with broadband down-converted light and entangled photons," Phys. Rev. A 76, 43813 (2007).
[CrossRef]

A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, "Temporal Shaping of Entangled Photons," Phys. Rev. Lett. 94, 073601 (2005).
[CrossRef] [PubMed]

Fei, H.

B. Saleh, B. Jost, H. Fei, and M. Teich, "Entangled-Photon Virtual-State Spectroscopy," Phys. Rev. Lett. 80, 3483 (1998).
[CrossRef]

H. Fei, B. Jost, S. Popescu, B. Saleh, and M. Teich, "Entanglement-Induced Two-Photon Transparency," Phys. Rev. Lett. 78, 1679-1682 (1997).
[CrossRef]

Friesem, A. A.

A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, "Temporal Shaping of Entangled Photons," Phys. Rev. Lett. 94, 073601 (2005).
[CrossRef] [PubMed]

Gasparoni, S.

P. Walther, J. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, "De Broglie wavelength of a nonlocal four-photon state," Nature 429, 158 (2004).
[CrossRef] [PubMed]

Glauber, R.

R. Glauber, "The photon theory of optical coherence," Phys. Rev. 130, 2529 (1963).
[CrossRef]

Goodson, T.

D. Lee and T. Goodson, "Entangled Photon Absorption in an Organic Porphyrin Dendrimer," J. Phys. Chem. B 110, 25582 (2006).
[CrossRef]

Gould, P.

J. Javanainen and P. Gould, "Linear intensity dependence of a two-photon transition rate," Phys. Rev. A 41, 5088 (1990).
[CrossRef] [PubMed]

Harbola, U.

C. Marx, U. Harbola, and S. Mukamel, "Nonlinear optical spectroscopy of single, few, and many molecules: Nonequilibrium Green’s function QED approach," Phys. Rev. A 77, 22110 (2008).
[CrossRef] [PubMed]

Hong, C. K.

C. K. Hong and L. Mandel, "Theory of parametric frequency down conversion of light," Phys. Rev. A 31, 2409 (1985).
[CrossRef] [PubMed]

Itano, W. M.

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, "Deterministic Entanglement of Two Trapped Ions," Phys. Rev. Lett. 81, 3631 (1998).
[CrossRef]

Javanainen, J.

J. Javanainen and P. Gould, "Linear intensity dependence of a two-photon transition rate," Phys. Rev. A 41, 5088 (1990).
[CrossRef] [PubMed]

Jost, B.

B. Saleh, B. Jost, H. Fei, and M. Teich, "Entangled-Photon Virtual-State Spectroscopy," Phys. Rev. Lett. 80, 3483 (1998).
[CrossRef]

H. Fei, B. Jost, S. Popescu, B. Saleh, and M. Teich, "Entanglement-Induced Two-Photon Transparency," Phys. Rev. Lett. 78, 1679-1682 (1997).
[CrossRef]

King, B. E.

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, "Deterministic Entanglement of Two Trapped Ions," Phys. Rev. Lett. 81, 3631 (1998).
[CrossRef]

Lee, D.

D. Lee and T. Goodson, "Entangled Photon Absorption in an Organic Porphyrin Dendrimer," J. Phys. Chem. B 110, 25582 (2006).
[CrossRef]

Leibfried, D.

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, "Deterministic Entanglement of Two Trapped Ions," Phys. Rev. Lett. 81, 3631 (1998).
[CrossRef]

Lissandrin, F.

F. Lissandrin, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Quantum theory of entangled-photon photoemission," Phys. Rev. B 69, 165317 (2004).
[CrossRef]

Mandel, L.

C. K. Hong and L. Mandel, "Theory of parametric frequency down conversion of light," Phys. Rev. A 31, 2409 (1985).
[CrossRef] [PubMed]

Marx, C.

O. Roslyak, C. Marx, and S. Mukamel, "Manipulating quantum pathways of matter with entangled photons," Phys. Rev. B (submitted) (2008).

O. Roslyak, C. Marx, and S. Mukamel, "A unified description of sum frequency generation, parametric down conversion and two photon fluorescence." J. Mol. Phys. (submitted) (2008).

C. Marx, U. Harbola, and S. Mukamel, "Nonlinear optical spectroscopy of single, few, and many molecules: Nonequilibrium Green’s function QED approach," Phys. Rev. A 77, 22110 (2008).
[CrossRef] [PubMed]

Monroe, C.

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, "Deterministic Entanglement of Two Trapped Ions," Phys. Rev. Lett. 81, 3631 (1998).
[CrossRef]

Mukamel, S.

O. Roslyak, C. Marx, and S. Mukamel, "A unified description of sum frequency generation, parametric down conversion and two photon fluorescence." J. Mol. Phys. (submitted) (2008).

C. Marx, U. Harbola, and S. Mukamel, "Nonlinear optical spectroscopy of single, few, and many molecules: Nonequilibrium Green’s function QED approach," Phys. Rev. A 77, 22110 (2008).
[CrossRef] [PubMed]

O. Roslyak, C. Marx, and S. Mukamel, "Manipulating quantum pathways of matter with entangled photons," Phys. Rev. B (submitted) (2008).

Myatt, C. J.

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, "Deterministic Entanglement of Two Trapped Ions," Phys. Rev. Lett. 81, 3631 (1998).
[CrossRef]

Nagasako, E. M.

E. M. Nagasako, S. J. Bentley, R. W. Boyd, and G. S. Agarwal, "Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers," Phys. Rev. A 64, 043802 (2001).
[CrossRef]

Pan, J.

P. Walther, J. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, "De Broglie wavelength of a nonlocal four-photon state," Nature 429, 158 (2004).
[CrossRef] [PubMed]

Pe’er, A.

A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, "Temporal Shaping of Entangled Photons," Phys. Rev. Lett. 94, 073601 (2005).
[CrossRef] [PubMed]

Popescu, S.

H. Fei, B. Jost, S. Popescu, B. Saleh, and M. Teich, "Entanglement-Induced Two-Photon Transparency," Phys. Rev. Lett. 78, 1679-1682 (1997).
[CrossRef]

Roslyak, O.

O. Roslyak, C. Marx, and S. Mukamel, "Manipulating quantum pathways of matter with entangled photons," Phys. Rev. B (submitted) (2008).

O. Roslyak, C. Marx, and S. Mukamel, "A unified description of sum frequency generation, parametric down conversion and two photon fluorescence." J. Mol. Phys. (submitted) (2008).

Saleh, B.

B. Saleh, B. Jost, H. Fei, and M. Teich, "Entangled-Photon Virtual-State Spectroscopy," Phys. Rev. Lett. 80, 3483 (1998).
[CrossRef]

H. Fei, B. Jost, S. Popescu, B. Saleh, and M. Teich, "Entanglement-Induced Two-Photon Transparency," Phys. Rev. Lett. 78, 1679-1682 (1997).
[CrossRef]

Saleh, B. E. A.

F. Lissandrin, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Quantum theory of entangled-photon photoemission," Phys. Rev. B 69, 165317 (2004).
[CrossRef]

Sergienko, A. V.

F. Lissandrin, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Quantum theory of entangled-photon photoemission," Phys. Rev. B 69, 165317 (2004).
[CrossRef]

Silberberg, Y.

A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, "Temporal Shaping of Entangled Photons," Phys. Rev. Lett. 94, 073601 (2005).
[CrossRef] [PubMed]

Teich, M.

B. Saleh, B. Jost, H. Fei, and M. Teich, "Entangled-Photon Virtual-State Spectroscopy," Phys. Rev. Lett. 80, 3483 (1998).
[CrossRef]

H. Fei, B. Jost, S. Popescu, B. Saleh, and M. Teich, "Entanglement-Induced Two-Photon Transparency," Phys. Rev. Lett. 78, 1679-1682 (1997).
[CrossRef]

Teich, M. C.

F. Lissandrin, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Quantum theory of entangled-photon photoemission," Phys. Rev. B 69, 165317 (2004).
[CrossRef]

Turchette, Q. A.

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, "Deterministic Entanglement of Two Trapped Ions," Phys. Rev. Lett. 81, 3631 (1998).
[CrossRef]

Ursin, R.

P. Walther, J. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, "De Broglie wavelength of a nonlocal four-photon state," Nature 429, 158 (2004).
[CrossRef] [PubMed]

Walther, P.

P. Walther, J. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, "De Broglie wavelength of a nonlocal four-photon state," Nature 429, 158 (2004).
[CrossRef] [PubMed]

Wineland, D. J.

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, "Deterministic Entanglement of Two Trapped Ions," Phys. Rev. Lett. 81, 3631 (1998).
[CrossRef]

Wood, C. S.

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, "Deterministic Entanglement of Two Trapped Ions," Phys. Rev. Lett. 81, 3631 (1998).
[CrossRef]

Zeilinger, A.

P. Walther, J. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, "De Broglie wavelength of a nonlocal four-photon state," Nature 429, 158 (2004).
[CrossRef] [PubMed]

J. Mol. Phys.

O. Roslyak, C. Marx, and S. Mukamel, "A unified description of sum frequency generation, parametric down conversion and two photon fluorescence." J. Mol. Phys. (submitted) (2008).

J. Phys. Chem. B

D. Lee and T. Goodson, "Entangled Photon Absorption in an Organic Porphyrin Dendrimer," J. Phys. Chem. B 110, 25582 (2006).
[CrossRef]

Nature

P. Walther, J. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, "De Broglie wavelength of a nonlocal four-photon state," Nature 429, 158 (2004).
[CrossRef] [PubMed]

Phys. Rev.

R. Glauber, "The photon theory of optical coherence," Phys. Rev. 130, 2529 (1963).
[CrossRef]

Phys. Rev. A

C. Marx, U. Harbola, and S. Mukamel, "Nonlinear optical spectroscopy of single, few, and many molecules: Nonequilibrium Green’s function QED approach," Phys. Rev. A 77, 22110 (2008).
[CrossRef] [PubMed]

J. Javanainen and P. Gould, "Linear intensity dependence of a two-photon transition rate," Phys. Rev. A 41, 5088 (1990).
[CrossRef] [PubMed]

B. Dayan, "Theory of two-photon interactions with broadband down-converted light and entangled photons," Phys. Rev. A 76, 43813 (2007).
[CrossRef]

C. K. Hong and L. Mandel, "Theory of parametric frequency down conversion of light," Phys. Rev. A 31, 2409 (1985).
[CrossRef] [PubMed]

E. M. Nagasako, S. J. Bentley, R. W. Boyd, and G. S. Agarwal, "Nonclassical two-photon interferometry and lithography with high-gain parametric amplifiers," Phys. Rev. A 64, 043802 (2001).
[CrossRef]

Phys. Rev. B

F. Lissandrin, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Quantum theory of entangled-photon photoemission," Phys. Rev. B 69, 165317 (2004).
[CrossRef]

O. Roslyak, C. Marx, and S. Mukamel, "Manipulating quantum pathways of matter with entangled photons," Phys. Rev. B (submitted) (2008).

Phys. Rev. Lett.

B. Saleh, B. Jost, H. Fei, and M. Teich, "Entangled-Photon Virtual-State Spectroscopy," Phys. Rev. Lett. 80, 3483 (1998).
[CrossRef]

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, C. Monroe, and D. J. Wineland, "Deterministic Entanglement of Two Trapped Ions," Phys. Rev. Lett. 81, 3631 (1998).
[CrossRef]

H. Fei, B. Jost, S. Popescu, B. Saleh, and M. Teich, "Entanglement-Induced Two-Photon Transparency," Phys. Rev. Lett. 78, 1679-1682 (1997).
[CrossRef]

A. Pe’er, B. Dayan, A. A. Friesem, and Y. Silberberg, "Temporal Shaping of Entangled Photons," Phys. Rev. Lett. 94, 073601 (2005).
[CrossRef] [PubMed]

Other

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University Press, 2005).

R. Glauber, Quantum Theory of Optical Coherence: Selected Papers and Lectures (Wiley-VCH, 2007).

M. Teich and B. Saleh, "Entangled-photon microscopy, spectroscopy, and display," US Patent 5,796,477 (1998).

S. Mukamel, Principles of nonlinear optical spectroscopy (Oxford University Press New York, 1995).

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Figures (6)

Fig. 1.
Fig. 1.

DFG: (A) wave-vector configuration of the optical fields corresponding to the phase matching k 3 = k 1k 2. (B) molecular level scheme. (C) Liouville space pathways for the pair of molecules contributing to the signal molecule a (C1,C2) and b (C1*,C2*)

Fig. 2.
Fig. 2.

(Color online) Optical field SNGF which contribute to the DFG process. Interactions with molecule a occur at times t 4, t 2 (red arrows), and with molecule b at times t 3, t 1(blue arrows) . Hilbert Space expressions for the signal are obtained by proceeding clockwise along the loop, starting at the bottom left.

Fig. 3.
Fig. 3.

Nonlinear spectroscopy with entangled photons. A non-linear parametric down conversion χ (2) crystal PDC is used to obtain entangled photon pairs from the classical pump beam by parametric down conversion. BS are balanced 50 : 50 beam splitters. ϕ is a phase shift in one of the interferometer arms. The sample is a collection of N three-level molecules. a1 ,a2 are annihilation operators for the incoming non-entangled (canonical) modes and a 1 ,a 2 represent the entangled modes.

Fig. 4.
Fig. 4.

(Color online) Panels (A-C) are 2D spectra of coherent DFG signals. (A) generated by classical fields, (B) generated by maximally entangled photons (PDC/MZI) in the low pump intensity limit. (C) generated by fields in a coherent state of low intensity. (D) the incoherent TPEF signal with classical k 1, k2 modes.

Fig. 5.
Fig. 5.

(A) 1D section of the 2D DFG spectra along of Fig.4 the line (d) in panels (A, dotted curve), (B, solid thick curve), (C, solid thin curve). (B) same as panel (A) but for a different section (the line (e)) in Fig. 4.

Fig. 6.
Fig. 6.

(A) The incoherent TPF pathway contributing at ω 3ω 1ω 2 resonance. Mode k 3 is spontaneously generated by classical modes k 1 ,k 2. (B) CTPL diagram for conventional incoherent two-photon emitted fluorescence (TPEF) signal, when mode k 3, k 2 are spontaneously generated by the classical mode k 1; (C) the loop diagram for two-photon induced fluorescence (TPIF) with classical k 1, k 2 modes; maximum of the signal corresponds to ω 3ω 1 + ω 2 .

Equations (58)

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

Hint=α=1,2,3Hα =Eα rt Vα, rt +c.c
Vα,rt=j=1Nδ(rRj)eiH0tVjα, eiH0t
Vjα,t=μgeαge +μefα e f+μgfα g f
Eαrt=2πωαΩeikαrαtaα
SHOM(ω3)=4πiω3Ωdr6dr5 a,baexp(ik3(r6r5)) ×
×dt6dt5eiω3(t6t5)VL3r6t6aVR3,r5t5bF
SDFG(ω3;ω2,ω1)=N(N1)iπω3Ω dt6 dt1 e3(t6t5) ×
[𝓣VL3(t6)VL2(t4)VL1,(t2)a𝓣VR3,(t5)VR2,(t3)VR1(t1)b𝓣EL2,(t4)EL1(t2)ER2(t3)ER1,(t1)+
𝓣VL3(t6)VR2(t4)VL1,(t2)a𝓣VR3,(t5)VL2,(t3)VR1(t1)b𝓣ER2,(t4)EL1(t2)EL2(t3)ER1,(t1)+
𝓣VL3(t6)VL2(t4)VL1,(t2)a𝓣VR3,(t5)VL2,(t3)VR1(t1)b𝓣EL2,(t4)EL1(t2)EL2(t3)ER1,(t1)+
𝓣VL3(t6)VR2(t4)VL1,(t2)a𝓣VR3,(t5)VR2,(t3)VR1(t1)b𝓣ER2,(t4)EL1(t2)ER2(t3)ER1,(t1)]
E2,(t4)E1(t2)E2(t3)E1,(t1)=𝓔12 𝓔22 exp (2(t4t3)2(t2t1))
SDFG(C)(ω3;ω2,ω1)=
=N(N1) (4πω3Ω)𝓔12𝓔22 χ+(2) (ω3;ω2,ω1)2 δ (ω3+ω2ω1)
χ+(2)(ω3;ω2,ω1)=12[χLLL(2)(ω3;ω2,ω1)+χLRL(2)(ω3;ω2,ω1)]
χv1v2v3(2)(ω3;ω2,ω1)=
21θ(τ3τ2)θ(τ3τ1) ei(ω2τ2ω1τ1) 𝓣Vv13(τ3)Vv22(τ2)Vv31,τ(τ1)
𝓣VL3(τ3)VL2(τ2)VL1,(τ1)=
=θ(τ2τ1)V3(τ3)V2(τ2)V1,(τ1)+θ(τ1τ2)V3(τ3)V1,(τ1)V2(τ2)
χLLL(2)(ω3;ω2,ω1)=12! μge μef μfg Ige (ω3)Ifg(ω1)
Iνν(ω)=1ωωνν+iγνν
χLRL(2)(ω3;ω2,ω1)=12! μge μef μfg Ige (ω2)Ifg(ω1)
SDFG(ω3;ω2,ω1)=
=N(N1)ω1 ω2 ω3 δ (ω3+ω2ω1) (Ω)3n=14Sn(ω3;ω2,ω1)
S1(ω3;ω2,ω1)=
χLLL(2) (ω3;ω2,ω1) χRRR(2) (ω3;ω2,ω1) a1a2a2a1
S2(ω3;ω2,ω1)=χLRL(2)(ω3;ω2,ω1) χRLR(2) ( ω3;ω2,ω1 ) ×
(a1a2a2a1+a1a2a1a2+a2a1a2a1+a2a1a1a2)
S3(ω3;ω2,ω1)=
χLLL(2)(ω3;ω2,ω1) χRLR(2) (ω3;ω2,ω1) (a1a2a2a1+a1a2a1a2a1a2a2a1)
S4(ω3;ω2,ω1)=
χLRL(2)(ω3;ω2,ω1) χRRR(2) (ω3;ω2,ω1) (a1a2a2a1+a1a2a2a1a2a1a2a1)
a1=12 [(1e)(Ua1+Va2)i(1+e)(Ua2+Va1)]
a2=12[i(1+e)(Ua1+Va2)(1e)(Ua'2+Va1)]
aaaa=V4(32+12cos2ϕ)
aaaa=V2[(12+12cos2ϕ)+V2(32+12cos2ϕ)]
S2(ω3;ω2,ω1)~4χLRL(2)(ω3;ω2,ω1)χRLR(2)(ω3;ω2,ω1)p2
S3(ω3;ω2,ω1)~2χLLL(2)(ω3;ω2,ω1)χRLR(2)(ω3;ω2,ω1)p2
S4(ω3;ω2,ω1)~2χLRL(2)(ω3;ω2,ω1)χRRR(2)(ω3;ω2,ω1)p2
SDFG(E)(ω1,ω2)~N(N1)p2×
[χLLL(2)(ω3;ω2,ω1)χRLR(2)(ω3;ω2,ω1)+χLRL(2)(ω3;ω2,ω1)χRLR(2)(ω3;ω2,ω1)]
SDFG(CS)(ω1,ω2)~N(N1)12×
[χRLR(2)(ω3;ω2,ω1)χLLL(2)(ω3;ω2,ω1)+χRRR(2)(ω3;ω2,ω1)χLLL(2)(ω3;ω2,ω1)]
SICOH(ω3)=4πiω3Ω d r6 d r5 a exp (ik3(r6r5))×
d t6 d t5 eiω3(t6t5) 𝓣VL(r6,t6)VR(r5,t5) a , F
STPF(ω3,ω2,ω1)=Nω3Ωdt6dt1eiω3(t6t5)×
𝓣VL3(t6)VL2(t4)VR1,(t2)VR3,(t5)VR2,(t3)VR1(t2) 𝓣EL2,(t4)EL1(t2)ER2(t3)ER1,(t1)
STPF(ω1,ω2)=NA πω3Ω χLLLRRR(5) (ω3;ω2,ω1,ω2,ω3,ω1)
χLLLRRR(5)(ω3;ω2,ω1,ω2,ω3,ω1)=
=15!μgeXμefXμfgY2δ(ω1ω2ω3ωgg')1ω1ωfg+iγfg1ω1ω2ωeg+iγeg2
S1(ω3;ω2,ω1)=
χLLL(2) ( ω3;ω2,ω1 ) χRRR(2) ( ω3;ω2,ω1 ) (β12+β12β22)
S2(ω3;ω2,ω1)=4χLRL(2)(ω3;ω2,ω1) χRLR(2) ( ω3;ω2,ω1 ) β12 β22
S3(ω3;ω2,ω1)~2χLLL(2)(ω3;ω2,ω1) χRLR(2) (ω3;ω2,ω1) (β12+3β12β22)
S4(ω3;ω2,ω1)~2χLRL(2)(ω3;ω2,ω1) χRRR(2) (ω3;ω2,ω1) (β12+3β12β22)
STPF(E)(ω1,ω2)~Nπω3ΩχLLLRRR(5)(ω3;ω2,ω1,ω2,ω3,ω1)
χLLLRRR(5)(ω3;ω2,ω1,ω2,ω3,ω1)=
=15!μgeXμefXμfgY2δ(ω1+ω2ω3ωgg)1ω1ωeg+iγfg1ω1+ω2ωfg+iγeg2

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