Abstract

We demonstrate the potential of semiconductor quantum dot nanomaterials for solid-state based controllable quantum memories in which losses may be compensated by gain. The dynamic photonic quantum-coherence present in a quantum dot ensemble and generated by a coherent signal pulse is influenced and controlled by disorder, spectral detuning and the power of the pulse. We show that the high coupling of spatial and temporal degrees of freedom is a key requirement for coherence transfer and/or storage.

© 2008 Optical Society of America

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  1. A. T. Black, J. K. Thompson and V. Vuletic, "On-Demand Superradiant Conversion of Atomic Spin Gratings into Single Photons with High Efficiency," Phys. Rev. Lett. 95, 133601 (2005).
    [CrossRef] [PubMed]
  2. M. D. Eisaman, L. Childress, A. Andre, F. Massou, A. S. Zibrov, and M. D. Lukin, "Shaping Quantum Pulses of Light Via Coherent Atomic Memory," Phys. Rev. Lett. 93, 233602 (2004).
    [CrossRef] [PubMed]
  3. A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, "Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles," Nature 423, 731-734 (2003).
    [CrossRef] [PubMed]
  4. C.W. Chou, S. V. Polyakov, A. Kuzmich, A, and H. J. Kimble, "Single-Photon Generation from Stored Excitation in an Atomic Ensemble." Phys. Rev. Lett. 92, 213601 (2004).
    [CrossRef] [PubMed]
  5. B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, "Experimental demonstration of quantum memory for light," Nature 432, 482-486 (2004).
    [CrossRef] [PubMed]
  6. D. N. Matsukevich and A. Kuzmich, "Quantum State transfer between Matter and Light," Science 306, 663-666 (2004).
    [CrossRef] [PubMed]
  7. N. S. Ginsberg, S. R. Garner, and L. V. Hau, "Coherent control of optical information with matter wave dynamics," Nature 445, 623-626 (2007).
    [CrossRef] [PubMed]
  8. K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "’Trapped rainbow’ storage of light in metamaterials," Nature 450, 397-401 (2007).
    [CrossRef] [PubMed]
  9. M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, "Optically programmable electron spin memory using semiconductor quantum dots," Nature 432, 81-84 (2004).
    [CrossRef] [PubMed]
  10. C. Santori, D. Fattal, D. J. Vuckovic, G. S. Solomon, and Y. Yamamoto, "Indistiguishable photons from a singlephoton device," Nature 419, 594-597 (2002).
    [CrossRef] [PubMed]
  11. E. Gehrig and O. Hess, Spatio-Temporal Dynamics and Quantum Fluctuations in Semiconductor Lasers (Springer, Heidelberg, 2003).
    [CrossRef]
  12. M. Kira, F. Jahnke, and S. W. Koch, "Microscopic Theory of Excitonic Signatures in Semiconductor Photoluminescence," Phys. Rev. Lett. 81, 3263-3266 (1998).
    [CrossRef]
  13. J. Fiurasek, "Optimal probabilistic estimation of quantum states," New J. Phys. 8, 192-199 (2006).
    [CrossRef]
  14. C. W. Gardiner and P. Zoller, Quantum Noise (Springer, Berlin, 2000).
  15. T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, "Storage and retrieval of single photons transmitted between remote quantum memories," Nature 438, 83 (2005).
    [CrossRef]

2007

N. S. Ginsberg, S. R. Garner, and L. V. Hau, "Coherent control of optical information with matter wave dynamics," Nature 445, 623-626 (2007).
[CrossRef] [PubMed]

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "’Trapped rainbow’ storage of light in metamaterials," Nature 450, 397-401 (2007).
[CrossRef] [PubMed]

2006

J. Fiurasek, "Optimal probabilistic estimation of quantum states," New J. Phys. 8, 192-199 (2006).
[CrossRef]

2005

T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, "Storage and retrieval of single photons transmitted between remote quantum memories," Nature 438, 83 (2005).
[CrossRef]

A. T. Black, J. K. Thompson and V. Vuletic, "On-Demand Superradiant Conversion of Atomic Spin Gratings into Single Photons with High Efficiency," Phys. Rev. Lett. 95, 133601 (2005).
[CrossRef] [PubMed]

2004

M. D. Eisaman, L. Childress, A. Andre, F. Massou, A. S. Zibrov, and M. D. Lukin, "Shaping Quantum Pulses of Light Via Coherent Atomic Memory," Phys. Rev. Lett. 93, 233602 (2004).
[CrossRef] [PubMed]

C.W. Chou, S. V. Polyakov, A. Kuzmich, A, and H. J. Kimble, "Single-Photon Generation from Stored Excitation in an Atomic Ensemble." Phys. Rev. Lett. 92, 213601 (2004).
[CrossRef] [PubMed]

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, "Experimental demonstration of quantum memory for light," Nature 432, 482-486 (2004).
[CrossRef] [PubMed]

D. N. Matsukevich and A. Kuzmich, "Quantum State transfer between Matter and Light," Science 306, 663-666 (2004).
[CrossRef] [PubMed]

M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, "Optically programmable electron spin memory using semiconductor quantum dots," Nature 432, 81-84 (2004).
[CrossRef] [PubMed]

2003

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, "Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles," Nature 423, 731-734 (2003).
[CrossRef] [PubMed]

2002

C. Santori, D. Fattal, D. J. Vuckovic, G. S. Solomon, and Y. Yamamoto, "Indistiguishable photons from a singlephoton device," Nature 419, 594-597 (2002).
[CrossRef] [PubMed]

1998

M. Kira, F. Jahnke, and S. W. Koch, "Microscopic Theory of Excitonic Signatures in Semiconductor Photoluminescence," Phys. Rev. Lett. 81, 3263-3266 (1998).
[CrossRef]

Abstreiter, G.

M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, "Optically programmable electron spin memory using semiconductor quantum dots," Nature 432, 81-84 (2004).
[CrossRef] [PubMed]

Andre, A.

M. D. Eisaman, L. Childress, A. Andre, F. Massou, A. S. Zibrov, and M. D. Lukin, "Shaping Quantum Pulses of Light Via Coherent Atomic Memory," Phys. Rev. Lett. 93, 233602 (2004).
[CrossRef] [PubMed]

Bichler, M.

M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, "Optically programmable electron spin memory using semiconductor quantum dots," Nature 432, 81-84 (2004).
[CrossRef] [PubMed]

Black, A. T.

A. T. Black, J. K. Thompson and V. Vuletic, "On-Demand Superradiant Conversion of Atomic Spin Gratings into Single Photons with High Efficiency," Phys. Rev. Lett. 95, 133601 (2005).
[CrossRef] [PubMed]

Boardman, A. D.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "’Trapped rainbow’ storage of light in metamaterials," Nature 450, 397-401 (2007).
[CrossRef] [PubMed]

Boca, A.

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, "Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles," Nature 423, 731-734 (2003).
[CrossRef] [PubMed]

Boozer, A. D.

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, "Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles," Nature 423, 731-734 (2003).
[CrossRef] [PubMed]

Bowen, W. P.

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, "Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles," Nature 423, 731-734 (2003).
[CrossRef] [PubMed]

Chaneliere, T.

T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, "Storage and retrieval of single photons transmitted between remote quantum memories," Nature 438, 83 (2005).
[CrossRef]

Childress, L.

M. D. Eisaman, L. Childress, A. Andre, F. Massou, A. S. Zibrov, and M. D. Lukin, "Shaping Quantum Pulses of Light Via Coherent Atomic Memory," Phys. Rev. Lett. 93, 233602 (2004).
[CrossRef] [PubMed]

Chou, C. W.

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, "Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles," Nature 423, 731-734 (2003).
[CrossRef] [PubMed]

Chou, C.W.

C.W. Chou, S. V. Polyakov, A. Kuzmich, A, and H. J. Kimble, "Single-Photon Generation from Stored Excitation in an Atomic Ensemble." Phys. Rev. Lett. 92, 213601 (2004).
[CrossRef] [PubMed]

Cirac, J. I.

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, "Experimental demonstration of quantum memory for light," Nature 432, 482-486 (2004).
[CrossRef] [PubMed]

Duan, L.-M.

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, "Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles," Nature 423, 731-734 (2003).
[CrossRef] [PubMed]

Ducommun, Y.

M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, "Optically programmable electron spin memory using semiconductor quantum dots," Nature 432, 81-84 (2004).
[CrossRef] [PubMed]

Eisaman, M. D.

M. D. Eisaman, L. Childress, A. Andre, F. Massou, A. S. Zibrov, and M. D. Lukin, "Shaping Quantum Pulses of Light Via Coherent Atomic Memory," Phys. Rev. Lett. 93, 233602 (2004).
[CrossRef] [PubMed]

Fattal, D.

C. Santori, D. Fattal, D. J. Vuckovic, G. S. Solomon, and Y. Yamamoto, "Indistiguishable photons from a singlephoton device," Nature 419, 594-597 (2002).
[CrossRef] [PubMed]

Finley, J. J.

M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, "Optically programmable electron spin memory using semiconductor quantum dots," Nature 432, 81-84 (2004).
[CrossRef] [PubMed]

Fiurasek, J.

J. Fiurasek, "Optimal probabilistic estimation of quantum states," New J. Phys. 8, 192-199 (2006).
[CrossRef]

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, "Experimental demonstration of quantum memory for light," Nature 432, 482-486 (2004).
[CrossRef] [PubMed]

Garner, S. R.

N. S. Ginsberg, S. R. Garner, and L. V. Hau, "Coherent control of optical information with matter wave dynamics," Nature 445, 623-626 (2007).
[CrossRef] [PubMed]

Ginsberg, N. S.

N. S. Ginsberg, S. R. Garner, and L. V. Hau, "Coherent control of optical information with matter wave dynamics," Nature 445, 623-626 (2007).
[CrossRef] [PubMed]

Hau, L. V.

N. S. Ginsberg, S. R. Garner, and L. V. Hau, "Coherent control of optical information with matter wave dynamics," Nature 445, 623-626 (2007).
[CrossRef] [PubMed]

Heiss, D.

M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, "Optically programmable electron spin memory using semiconductor quantum dots," Nature 432, 81-84 (2004).
[CrossRef] [PubMed]

Hess, O.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "’Trapped rainbow’ storage of light in metamaterials," Nature 450, 397-401 (2007).
[CrossRef] [PubMed]

Jahnke, F.

M. Kira, F. Jahnke, and S. W. Koch, "Microscopic Theory of Excitonic Signatures in Semiconductor Photoluminescence," Phys. Rev. Lett. 81, 3263-3266 (1998).
[CrossRef]

Jenkins, S. D.

T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, "Storage and retrieval of single photons transmitted between remote quantum memories," Nature 438, 83 (2005).
[CrossRef]

Julsgaard, B.

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, "Experimental demonstration of quantum memory for light," Nature 432, 482-486 (2004).
[CrossRef] [PubMed]

Kennedy, T. A. B.

T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, "Storage and retrieval of single photons transmitted between remote quantum memories," Nature 438, 83 (2005).
[CrossRef]

Kimble, H. J.

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, "Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles," Nature 423, 731-734 (2003).
[CrossRef] [PubMed]

Kira, M.

M. Kira, F. Jahnke, and S. W. Koch, "Microscopic Theory of Excitonic Signatures in Semiconductor Photoluminescence," Phys. Rev. Lett. 81, 3263-3266 (1998).
[CrossRef]

Koch, S. W.

M. Kira, F. Jahnke, and S. W. Koch, "Microscopic Theory of Excitonic Signatures in Semiconductor Photoluminescence," Phys. Rev. Lett. 81, 3263-3266 (1998).
[CrossRef]

Kroutvar, M.

M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, "Optically programmable electron spin memory using semiconductor quantum dots," Nature 432, 81-84 (2004).
[CrossRef] [PubMed]

Kuzmich, A.

T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, "Storage and retrieval of single photons transmitted between remote quantum memories," Nature 438, 83 (2005).
[CrossRef]

C.W. Chou, S. V. Polyakov, A. Kuzmich, A, and H. J. Kimble, "Single-Photon Generation from Stored Excitation in an Atomic Ensemble." Phys. Rev. Lett. 92, 213601 (2004).
[CrossRef] [PubMed]

D. N. Matsukevich and A. Kuzmich, "Quantum State transfer between Matter and Light," Science 306, 663-666 (2004).
[CrossRef] [PubMed]

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, "Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles," Nature 423, 731-734 (2003).
[CrossRef] [PubMed]

Lan, S.-Y.

T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, "Storage and retrieval of single photons transmitted between remote quantum memories," Nature 438, 83 (2005).
[CrossRef]

Lukin, M. D.

M. D. Eisaman, L. Childress, A. Andre, F. Massou, A. S. Zibrov, and M. D. Lukin, "Shaping Quantum Pulses of Light Via Coherent Atomic Memory," Phys. Rev. Lett. 93, 233602 (2004).
[CrossRef] [PubMed]

Massou, F.

M. D. Eisaman, L. Childress, A. Andre, F. Massou, A. S. Zibrov, and M. D. Lukin, "Shaping Quantum Pulses of Light Via Coherent Atomic Memory," Phys. Rev. Lett. 93, 233602 (2004).
[CrossRef] [PubMed]

Matsukevich, D. N.

T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, "Storage and retrieval of single photons transmitted between remote quantum memories," Nature 438, 83 (2005).
[CrossRef]

D. N. Matsukevich and A. Kuzmich, "Quantum State transfer between Matter and Light," Science 306, 663-666 (2004).
[CrossRef] [PubMed]

Polyakov, S. V.

C.W. Chou, S. V. Polyakov, A. Kuzmich, A, and H. J. Kimble, "Single-Photon Generation from Stored Excitation in an Atomic Ensemble." Phys. Rev. Lett. 92, 213601 (2004).
[CrossRef] [PubMed]

Polzik, E. S.

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, "Experimental demonstration of quantum memory for light," Nature 432, 482-486 (2004).
[CrossRef] [PubMed]

Santori, C.

C. Santori, D. Fattal, D. J. Vuckovic, G. S. Solomon, and Y. Yamamoto, "Indistiguishable photons from a singlephoton device," Nature 419, 594-597 (2002).
[CrossRef] [PubMed]

Schuh, D.

M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, "Optically programmable electron spin memory using semiconductor quantum dots," Nature 432, 81-84 (2004).
[CrossRef] [PubMed]

Sherson, J.

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, "Experimental demonstration of quantum memory for light," Nature 432, 482-486 (2004).
[CrossRef] [PubMed]

Thompson, J. K.

A. T. Black, J. K. Thompson and V. Vuletic, "On-Demand Superradiant Conversion of Atomic Spin Gratings into Single Photons with High Efficiency," Phys. Rev. Lett. 95, 133601 (2005).
[CrossRef] [PubMed]

Tsakmakidis, K. L.

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "’Trapped rainbow’ storage of light in metamaterials," Nature 450, 397-401 (2007).
[CrossRef] [PubMed]

Vuletic, V.

A. T. Black, J. K. Thompson and V. Vuletic, "On-Demand Superradiant Conversion of Atomic Spin Gratings into Single Photons with High Efficiency," Phys. Rev. Lett. 95, 133601 (2005).
[CrossRef] [PubMed]

Zibrov, A. S.

M. D. Eisaman, L. Childress, A. Andre, F. Massou, A. S. Zibrov, and M. D. Lukin, "Shaping Quantum Pulses of Light Via Coherent Atomic Memory," Phys. Rev. Lett. 93, 233602 (2004).
[CrossRef] [PubMed]

Nature

A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L.-M. Duan, and H. J. Kimble, "Generation of nonclassical photon pairs for scalable quantum communication with atomic ensembles," Nature 423, 731-734 (2003).
[CrossRef] [PubMed]

N. S. Ginsberg, S. R. Garner, and L. V. Hau, "Coherent control of optical information with matter wave dynamics," Nature 445, 623-626 (2007).
[CrossRef] [PubMed]

K. L. Tsakmakidis, A. D. Boardman, and O. Hess, "’Trapped rainbow’ storage of light in metamaterials," Nature 450, 397-401 (2007).
[CrossRef] [PubMed]

M. Kroutvar, Y. Ducommun, D. Heiss, M. Bichler, D. Schuh, G. Abstreiter, and J. J. Finley, "Optically programmable electron spin memory using semiconductor quantum dots," Nature 432, 81-84 (2004).
[CrossRef] [PubMed]

C. Santori, D. Fattal, D. J. Vuckovic, G. S. Solomon, and Y. Yamamoto, "Indistiguishable photons from a singlephoton device," Nature 419, 594-597 (2002).
[CrossRef] [PubMed]

B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, "Experimental demonstration of quantum memory for light," Nature 432, 482-486 (2004).
[CrossRef] [PubMed]

T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, "Storage and retrieval of single photons transmitted between remote quantum memories," Nature 438, 83 (2005).
[CrossRef]

New J. Phys.

J. Fiurasek, "Optimal probabilistic estimation of quantum states," New J. Phys. 8, 192-199 (2006).
[CrossRef]

Phys. Rev. Lett.

M. Kira, F. Jahnke, and S. W. Koch, "Microscopic Theory of Excitonic Signatures in Semiconductor Photoluminescence," Phys. Rev. Lett. 81, 3263-3266 (1998).
[CrossRef]

C.W. Chou, S. V. Polyakov, A. Kuzmich, A, and H. J. Kimble, "Single-Photon Generation from Stored Excitation in an Atomic Ensemble." Phys. Rev. Lett. 92, 213601 (2004).
[CrossRef] [PubMed]

A. T. Black, J. K. Thompson and V. Vuletic, "On-Demand Superradiant Conversion of Atomic Spin Gratings into Single Photons with High Efficiency," Phys. Rev. Lett. 95, 133601 (2005).
[CrossRef] [PubMed]

M. D. Eisaman, L. Childress, A. Andre, F. Massou, A. S. Zibrov, and M. D. Lukin, "Shaping Quantum Pulses of Light Via Coherent Atomic Memory," Phys. Rev. Lett. 93, 233602 (2004).
[CrossRef] [PubMed]

Science

D. N. Matsukevich and A. Kuzmich, "Quantum State transfer between Matter and Light," Science 306, 663-666 (2004).
[CrossRef] [PubMed]

Other

C. W. Gardiner and P. Zoller, Quantum Noise (Springer, Berlin, 2000).

E. Gehrig and O. Hess, Spatio-Temporal Dynamics and Quantum Fluctuations in Semiconductor Lasers (Springer, Heidelberg, 2003).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Snapshots of the carrier density (a), field-field correlation (b) and field-dipole correlation (c) in the optically excited quantum dot ensemble.

Fig. 2.
Fig. 2.

Transfer (a) and storage (b) of information. Field-dipole and field-field FOM in the centre (left) and at the rim (right) of the QD medium: (a) Resonant injection, (b) Off-resonant injection.

Fig. 3.
Fig. 3.

Longitudinal cuts of Re(Ieh ) (top row) and Im(Ceh ) (bottom row). The spectral detuning is (a) 6 nm and (b) 3 nm, (c) refers to resonant injection.

Fig. 4.
Fig. 4.

Influence of disorder: Snapshots of the carrier density (a), field-field correlation (b) and field-dipole correlation (c) for low (left) and high (right) excitation strength (in units of the saturation power Ps .)

Equations (9)

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t I ij ( r , r , k e , k h ) = ( κ i + κ j ) I ij ( r , r , k e , k h )
i ω 0 2 k 0 2 ε r ( Δ r Δ r ) I ij ( r , r , k e , k h )
i g 0 v 0 4 π 2 [ C ij ( r , r , k e , k h ) C ji * ( r , r , k e , k h )
+ 1 k 0 2 ε r ( 2 r j r k C ik ( r , r , k e , k h ) 2 r i r k C jk * ( r , r , k e , k h ) ) ]
t C ij ( r , r , k e , k h ) = [ Γ k e , k h i Ω k e , k h ] C ij ( r , r , k e , k h )
( κ i i δ ω i ) C ij ( r , r , k e , k h ) i ω 0 2 k 0 2 ε r Δ r C ij ( r , r , k e , k h )
+ i g 0 σ v 0 ( f e ( r , k e , k h ) ) + f h ( r , k e , k h ) 1 ) U ij ( r , r , k e , k h )
+ i g 0 σ v 0 δ ( r r ) δ ij ( f e ( r , k e , k h ) · f h ( r , k e , k h )
+ p ( r , k e , k h ) · p * ( r , k e , k h ) )

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