Abstract

A fully quantized analysis is presented on the origin of induced magnetic dipole (MD) scattering in two-level diatomic molecules. The interaction is driven by dual optical fields, E and H*, and is universally allowed in dielectric optical materials, including centrosymmetric media. Leading terms of the interaction are shown to be quadratic and cubic with respect to the intensity, predicting an upper limit for the induced magnetic dipole scattering intensity (IMDm2) that is equal to the electric dipole scattering (IEDp2). The optical dynamics proceed by first establishing an electric polarization in the system. Then the magnetic field exerts torque on the orbital angular momentum of the excited state, mediating an exchange of orbital and rotational angular momenta that enhances the magnetic moment. The magneto-electric interaction also accounts for second-order, unpolarized scattering from high-frequency librations previously ascribed to third-order, all-electric processes.

© 2016 Optical Society of America

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References

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  1. J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108(9), 097402 (2012).
    [Crossref] [PubMed]
  2. W. F. Koehl, B. B. Buckley, F. J. Heremans, G. Calusine, and D. D. Awschalom, “Room temperature coherent control of defect spin qubits in silicon carbide,” Nature 479(7371), 84–87 (2011).
    [Crossref] [PubMed]
  3. G. D. Fuchs, G. Burkard, P. V. Klimov, and D. D. Awschalom, “A quantum memory intrinsic to single nitrogen–vacancy centres in diamond,” Nat. Phys. 7(10), 789–793 (2011).
    [Crossref]
  4. C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, “All-optical magnetic recording with circularly polarized light,” Phys. Rev. Lett. 99(4), 047601 (2007).
    [Crossref] [PubMed]
  5. W. M. Fisher and S. C. Rand, “Optically-induced charge separation and terahertz emission in unbiased dielectrics,” J. Appl. Phys. 109(6), 064903 (2011).
    [Crossref]
  6. S. L. Oliveira and S. C. Rand, “Intense nonlinear magnetic dipole radiation at optical frequencies: molecular scattering in a dielectric liquid,” Phys. Rev. Lett. 98(9), 093901 (2007).
    [Crossref] [PubMed]
  7. S. C. Rand, W. M. Fisher, and S. L. Oliveira, “Optically induced magnetization in homogeneous, undoped dielectric media,” J. Opt. Soc. Am. B 25(7), 1106 (2008).
    [Crossref]
  8. W. M. Fisher and S. C. Rand, “Dependence of optically induced magnetism on molecular electronic structure,” J. Lumin. 129(12), 1407–1409 (2009).
    [Crossref]
  9. K. Y. Bliokh, Y. S. Kivshar, and F. Nori, “Magnetoelectric effects in local light-matter interactions,” Phys. Rev. Lett. 113(3), 033601 (2014).
    [Crossref] [PubMed]
  10. A. Einstein and W. J. de Haas, “Experimental proof of the existence of Ampère’s molecular currents,” K. Akad. van Wet. Amsterdam, Proc. 18, 696–711 (1915).
  11. A. A. Fisher, E. F. C. Dreyer, A. Chakrabarty, and S. C. Rand, “Optical magnetization, Part I: Experiments on radiant optical magnetization in solids,” Opt. Express 24(23), 26055–26063 (2016).
  12. N. A. Spaldin and M. Fiebig, “Materials science. The renaissance of magnetoelectric multiferroics,” Science 309(5733), 391–392 (2005).
    [Crossref] [PubMed]
  13. V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics, 2nd ed. (Butterworth-Heinemann, 1982), Vol. 4.
  14. A. A. Fisher, E. F. Cloos, W. M. Fisher, and S. C. Rand, “Dynamic symmetry-breaking in a simple quantum model of magneto-electric rectification, optical magnetization, and harmonic generation,” Opt. Express 22(3), 2910–2924 (2014).
    [Crossref] [PubMed]
  15. G. Herzberg, Spectra of Diatomic Molecules, 2nd ed. (Van Nostrand Reinhold, 1950).
  16. C. Cohen-Tannoudji and S. Reynaud, “Dressed Atom Approach to Resonance Fluorescence,” in Multiphoton Processes, J. Eberly and P. Lambropoulos, eds. (J. Wiley & Sons Inc., 1977), pp. 103–118.
  17. S. C. Rand, Lectures on Light, 2nd ed. (Oxford University Press, 2016).
  18. C. H. Townes and A. L. Schawlow, Microwave Spectroscopy (Dover, 1975).

2016 (1)

2014 (2)

2012 (1)

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108(9), 097402 (2012).
[Crossref] [PubMed]

2011 (3)

W. F. Koehl, B. B. Buckley, F. J. Heremans, G. Calusine, and D. D. Awschalom, “Room temperature coherent control of defect spin qubits in silicon carbide,” Nature 479(7371), 84–87 (2011).
[Crossref] [PubMed]

G. D. Fuchs, G. Burkard, P. V. Klimov, and D. D. Awschalom, “A quantum memory intrinsic to single nitrogen–vacancy centres in diamond,” Nat. Phys. 7(10), 789–793 (2011).
[Crossref]

W. M. Fisher and S. C. Rand, “Optically-induced charge separation and terahertz emission in unbiased dielectrics,” J. Appl. Phys. 109(6), 064903 (2011).
[Crossref]

2009 (1)

W. M. Fisher and S. C. Rand, “Dependence of optically induced magnetism on molecular electronic structure,” J. Lumin. 129(12), 1407–1409 (2009).
[Crossref]

2008 (1)

2007 (2)

S. L. Oliveira and S. C. Rand, “Intense nonlinear magnetic dipole radiation at optical frequencies: molecular scattering in a dielectric liquid,” Phys. Rev. Lett. 98(9), 093901 (2007).
[Crossref] [PubMed]

C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, “All-optical magnetic recording with circularly polarized light,” Phys. Rev. Lett. 99(4), 047601 (2007).
[Crossref] [PubMed]

2005 (1)

N. A. Spaldin and M. Fiebig, “Materials science. The renaissance of magnetoelectric multiferroics,” Science 309(5733), 391–392 (2005).
[Crossref] [PubMed]

1915 (1)

A. Einstein and W. J. de Haas, “Experimental proof of the existence of Ampère’s molecular currents,” K. Akad. van Wet. Amsterdam, Proc. 18, 696–711 (1915).

Awschalom, D. D.

W. F. Koehl, B. B. Buckley, F. J. Heremans, G. Calusine, and D. D. Awschalom, “Room temperature coherent control of defect spin qubits in silicon carbide,” Nature 479(7371), 84–87 (2011).
[Crossref] [PubMed]

G. D. Fuchs, G. Burkard, P. V. Klimov, and D. D. Awschalom, “A quantum memory intrinsic to single nitrogen–vacancy centres in diamond,” Nat. Phys. 7(10), 789–793 (2011).
[Crossref]

Basilio, L. I.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108(9), 097402 (2012).
[Crossref] [PubMed]

Bliokh, K. Y.

K. Y. Bliokh, Y. S. Kivshar, and F. Nori, “Magnetoelectric effects in local light-matter interactions,” Phys. Rev. Lett. 113(3), 033601 (2014).
[Crossref] [PubMed]

Brener, I.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108(9), 097402 (2012).
[Crossref] [PubMed]

Buckley, B. B.

W. F. Koehl, B. B. Buckley, F. J. Heremans, G. Calusine, and D. D. Awschalom, “Room temperature coherent control of defect spin qubits in silicon carbide,” Nature 479(7371), 84–87 (2011).
[Crossref] [PubMed]

Burkard, G.

G. D. Fuchs, G. Burkard, P. V. Klimov, and D. D. Awschalom, “A quantum memory intrinsic to single nitrogen–vacancy centres in diamond,” Nat. Phys. 7(10), 789–793 (2011).
[Crossref]

Calusine, G.

W. F. Koehl, B. B. Buckley, F. J. Heremans, G. Calusine, and D. D. Awschalom, “Room temperature coherent control of defect spin qubits in silicon carbide,” Nature 479(7371), 84–87 (2011).
[Crossref] [PubMed]

Chakrabarty, A.

Clem, P. G.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108(9), 097402 (2012).
[Crossref] [PubMed]

Cloos, E. F.

de Haas, W. J.

A. Einstein and W. J. de Haas, “Experimental proof of the existence of Ampère’s molecular currents,” K. Akad. van Wet. Amsterdam, Proc. 18, 696–711 (1915).

Dreyer, E. F. C.

Einstein, A.

A. Einstein and W. J. de Haas, “Experimental proof of the existence of Ampère’s molecular currents,” K. Akad. van Wet. Amsterdam, Proc. 18, 696–711 (1915).

Fiebig, M.

N. A. Spaldin and M. Fiebig, “Materials science. The renaissance of magnetoelectric multiferroics,” Science 309(5733), 391–392 (2005).
[Crossref] [PubMed]

Fisher, A. A.

Fisher, W. M.

A. A. Fisher, E. F. Cloos, W. M. Fisher, and S. C. Rand, “Dynamic symmetry-breaking in a simple quantum model of magneto-electric rectification, optical magnetization, and harmonic generation,” Opt. Express 22(3), 2910–2924 (2014).
[Crossref] [PubMed]

W. M. Fisher and S. C. Rand, “Optically-induced charge separation and terahertz emission in unbiased dielectrics,” J. Appl. Phys. 109(6), 064903 (2011).
[Crossref]

W. M. Fisher and S. C. Rand, “Dependence of optically induced magnetism on molecular electronic structure,” J. Lumin. 129(12), 1407–1409 (2009).
[Crossref]

S. C. Rand, W. M. Fisher, and S. L. Oliveira, “Optically induced magnetization in homogeneous, undoped dielectric media,” J. Opt. Soc. Am. B 25(7), 1106 (2008).
[Crossref]

Fuchs, G. D.

G. D. Fuchs, G. Burkard, P. V. Klimov, and D. D. Awschalom, “A quantum memory intrinsic to single nitrogen–vacancy centres in diamond,” Nat. Phys. 7(10), 789–793 (2011).
[Crossref]

Ginn, J. C.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108(9), 097402 (2012).
[Crossref] [PubMed]

Hansteen, F.

C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, “All-optical magnetic recording with circularly polarized light,” Phys. Rev. Lett. 99(4), 047601 (2007).
[Crossref] [PubMed]

Heremans, F. J.

W. F. Koehl, B. B. Buckley, F. J. Heremans, G. Calusine, and D. D. Awschalom, “Room temperature coherent control of defect spin qubits in silicon carbide,” Nature 479(7371), 84–87 (2011).
[Crossref] [PubMed]

Hines, P. F.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108(9), 097402 (2012).
[Crossref] [PubMed]

Ihlefeld, J. F.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108(9), 097402 (2012).
[Crossref] [PubMed]

Itoh, A.

C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, “All-optical magnetic recording with circularly polarized light,” Phys. Rev. Lett. 99(4), 047601 (2007).
[Crossref] [PubMed]

Kimel, A. V.

C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, “All-optical magnetic recording with circularly polarized light,” Phys. Rev. Lett. 99(4), 047601 (2007).
[Crossref] [PubMed]

Kirilyuk, A.

C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, “All-optical magnetic recording with circularly polarized light,” Phys. Rev. Lett. 99(4), 047601 (2007).
[Crossref] [PubMed]

Kivshar, Y. S.

K. Y. Bliokh, Y. S. Kivshar, and F. Nori, “Magnetoelectric effects in local light-matter interactions,” Phys. Rev. Lett. 113(3), 033601 (2014).
[Crossref] [PubMed]

Klimov, P. V.

G. D. Fuchs, G. Burkard, P. V. Klimov, and D. D. Awschalom, “A quantum memory intrinsic to single nitrogen–vacancy centres in diamond,” Nat. Phys. 7(10), 789–793 (2011).
[Crossref]

Koehl, W. F.

W. F. Koehl, B. B. Buckley, F. J. Heremans, G. Calusine, and D. D. Awschalom, “Room temperature coherent control of defect spin qubits in silicon carbide,” Nature 479(7371), 84–87 (2011).
[Crossref] [PubMed]

Nori, F.

K. Y. Bliokh, Y. S. Kivshar, and F. Nori, “Magnetoelectric effects in local light-matter interactions,” Phys. Rev. Lett. 113(3), 033601 (2014).
[Crossref] [PubMed]

Oliveira, S. L.

S. C. Rand, W. M. Fisher, and S. L. Oliveira, “Optically induced magnetization in homogeneous, undoped dielectric media,” J. Opt. Soc. Am. B 25(7), 1106 (2008).
[Crossref]

S. L. Oliveira and S. C. Rand, “Intense nonlinear magnetic dipole radiation at optical frequencies: molecular scattering in a dielectric liquid,” Phys. Rev. Lett. 98(9), 093901 (2007).
[Crossref] [PubMed]

Peters, D. W.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108(9), 097402 (2012).
[Crossref] [PubMed]

Rand, S. C.

A. A. Fisher, E. F. C. Dreyer, A. Chakrabarty, and S. C. Rand, “Optical magnetization, Part I: Experiments on radiant optical magnetization in solids,” Opt. Express 24(23), 26055–26063 (2016).

A. A. Fisher, E. F. Cloos, W. M. Fisher, and S. C. Rand, “Dynamic symmetry-breaking in a simple quantum model of magneto-electric rectification, optical magnetization, and harmonic generation,” Opt. Express 22(3), 2910–2924 (2014).
[Crossref] [PubMed]

W. M. Fisher and S. C. Rand, “Optically-induced charge separation and terahertz emission in unbiased dielectrics,” J. Appl. Phys. 109(6), 064903 (2011).
[Crossref]

W. M. Fisher and S. C. Rand, “Dependence of optically induced magnetism on molecular electronic structure,” J. Lumin. 129(12), 1407–1409 (2009).
[Crossref]

S. C. Rand, W. M. Fisher, and S. L. Oliveira, “Optically induced magnetization in homogeneous, undoped dielectric media,” J. Opt. Soc. Am. B 25(7), 1106 (2008).
[Crossref]

S. L. Oliveira and S. C. Rand, “Intense nonlinear magnetic dipole radiation at optical frequencies: molecular scattering in a dielectric liquid,” Phys. Rev. Lett. 98(9), 093901 (2007).
[Crossref] [PubMed]

Rasing, T.

C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, “All-optical magnetic recording with circularly polarized light,” Phys. Rev. Lett. 99(4), 047601 (2007).
[Crossref] [PubMed]

Sinclair, M. B.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108(9), 097402 (2012).
[Crossref] [PubMed]

Spaldin, N. A.

N. A. Spaldin and M. Fiebig, “Materials science. The renaissance of magnetoelectric multiferroics,” Science 309(5733), 391–392 (2005).
[Crossref] [PubMed]

Stanciu, C. D.

C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, “All-optical magnetic recording with circularly polarized light,” Phys. Rev. Lett. 99(4), 047601 (2007).
[Crossref] [PubMed]

Stevens, J. O.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108(9), 097402 (2012).
[Crossref] [PubMed]

Tsukamoto, A.

C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, “All-optical magnetic recording with circularly polarized light,” Phys. Rev. Lett. 99(4), 047601 (2007).
[Crossref] [PubMed]

Warne, L. K.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108(9), 097402 (2012).
[Crossref] [PubMed]

Wendt, J. R.

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108(9), 097402 (2012).
[Crossref] [PubMed]

J. Appl. Phys. (1)

W. M. Fisher and S. C. Rand, “Optically-induced charge separation and terahertz emission in unbiased dielectrics,” J. Appl. Phys. 109(6), 064903 (2011).
[Crossref]

J. Lumin. (1)

W. M. Fisher and S. C. Rand, “Dependence of optically induced magnetism on molecular electronic structure,” J. Lumin. 129(12), 1407–1409 (2009).
[Crossref]

J. Opt. Soc. Am. B (1)

K. Akad. van Wet. Amsterdam, Proc. (1)

A. Einstein and W. J. de Haas, “Experimental proof of the existence of Ampère’s molecular currents,” K. Akad. van Wet. Amsterdam, Proc. 18, 696–711 (1915).

Nat. Phys. (1)

G. D. Fuchs, G. Burkard, P. V. Klimov, and D. D. Awschalom, “A quantum memory intrinsic to single nitrogen–vacancy centres in diamond,” Nat. Phys. 7(10), 789–793 (2011).
[Crossref]

Nature (1)

W. F. Koehl, B. B. Buckley, F. J. Heremans, G. Calusine, and D. D. Awschalom, “Room temperature coherent control of defect spin qubits in silicon carbide,” Nature 479(7371), 84–87 (2011).
[Crossref] [PubMed]

Opt. Express (2)

Phys. Rev. Lett. (4)

J. C. Ginn, I. Brener, D. W. Peters, J. R. Wendt, J. O. Stevens, P. F. Hines, L. I. Basilio, L. K. Warne, J. F. Ihlefeld, P. G. Clem, and M. B. Sinclair, “Realizing optical magnetism from dielectric metamaterials,” Phys. Rev. Lett. 108(9), 097402 (2012).
[Crossref] [PubMed]

C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, “All-optical magnetic recording with circularly polarized light,” Phys. Rev. Lett. 99(4), 047601 (2007).
[Crossref] [PubMed]

K. Y. Bliokh, Y. S. Kivshar, and F. Nori, “Magnetoelectric effects in local light-matter interactions,” Phys. Rev. Lett. 113(3), 033601 (2014).
[Crossref] [PubMed]

S. L. Oliveira and S. C. Rand, “Intense nonlinear magnetic dipole radiation at optical frequencies: molecular scattering in a dielectric liquid,” Phys. Rev. Lett. 98(9), 093901 (2007).
[Crossref] [PubMed]

Science (1)

N. A. Spaldin and M. Fiebig, “Materials science. The renaissance of magnetoelectric multiferroics,” Science 309(5733), 391–392 (2005).
[Crossref] [PubMed]

Other (5)

V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Quantum Electrodynamics, 2nd ed. (Butterworth-Heinemann, 1982), Vol. 4.

G. Herzberg, Spectra of Diatomic Molecules, 2nd ed. (Van Nostrand Reinhold, 1950).

C. Cohen-Tannoudji and S. Reynaud, “Dressed Atom Approach to Resonance Fluorescence,” in Multiphoton Processes, J. Eberly and P. Lambropoulos, eds. (J. Wiley & Sons Inc., 1977), pp. 103–118.

S. C. Rand, Lectures on Light, 2nd ed. (Oxford University Press, 2016).

C. H. Townes and A. L. Schawlow, Microwave Spectroscopy (Dover, 1975).

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Figures (5)

Fig. 1
Fig. 1

Energy levels of the molecular model showing the 2-photon transition (solid arrows) driven by the optical Eand B * fields The dashed downward arrow depicts a magnetic de-excitation channel that becomes an option if the excitation bandwidth exceeds ω φ .

Fig. 2
Fig. 2

Dressed state picture of three dipole moments formed by strong excitation of a nominally 2-level molecule during a 2-photon E B * process. p (1) (ω) x ^ is the linear ED polarization along the quantization axis. p (2) (0) z ^ and m (2) (ω) y ^ are nonlinear rectification and magnetization moments oriented along z ^ and y ^ respectively.

Fig. 3
Fig. 3

Squared values of the total magnetic moment and the first order electric dipole moment versus the number of incident photons in the (a) 3-state model and (b) the 4-state model. In both figures separate curves are shown for m ^ 2 with rotational frequencies (left to right) of ω φ / ω 0 = 10 7 , 10 5 , 10 3 .

Fig. 4
Fig. 4

Orbital angular momentum L of an electron about the internuclear axis of a diatomic molecule, visualized in cylindrical coordinates referenced to the center of mass (COM) and fixed in the molecule. No magnetic torque has been exerted on the system. Angular momentum is determined by coordinate r of the electron.

Fig. 5
Fig. 5

After the application of magnetic torque, electron motion is in a plane orthogonal to that in Fig. 4. It consists of rotation about an axis perpendicular to the internuclear axis, normal to the plane of the drawing. Angular momentum is determined chiefly by coordinate h of the electron since we assume h>>r .

Equations (63)

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

H ^ mf = H ^ mol + H ^ field =( ω 0 /2) σ ^ z + O ^ 2 /2I+ω a ^ + a ^ ,
E 1 = ω 0 2 +nω,
E 2 = E 1 +Δ= ω 0 2 +(n1)ω,
E 3 = E 2 Δ+ ω ϕ = ω 0 2 +nω+ ω ϕ .
H ^ int = H ^ int (e) + H ^ int (m) =g( σ ^ + a ^ +h.c.)+(f L ^ ' O ^ ' + a ^ + +h.c.).
H=( E 3 2f n 0 2 f * n E 2 g n 0 g * n E 1 ).
( H E Di I )| D i (n) =0,
| E 3 E Di 2f n 0 2 f * n E 2 E Di g n 0 g * n E 1 E Di |=0
y 3 +p y 2 +qy+r=0,
p( E 1 + E 2 + E 3 )
q( E 1 E 2 + E 2 E 3 + E 3 E 1 4n 2 | f | 2 n 2 g 2 )
r( E 1 E 2 E 3 +4n 2 | f | 2 E 1 +n 2 g 2 E 3 ).
x 3 +ax+b=0,
a= 1 3 (3q p 2 ) =( E 1 E 2 + E 2 E 3 + E 3 E 1 4n 2 | f | 2 n 2 g 2 ) 1 3 ( E 1 + E 2 + E 3 ) 2
b= 1 27 (2 p 3 +27r9pq) = 2 27 ( E 1 + E 2 + E 3 ) 3 +( E 1 E 2 E 3 +4n 2 | f | 2 E 1 +n 2 g 2 E 3 ) + 1 3 ( E 1 + E 2 + E 3 )( E 1 E 2 + E 2 E 3 + E 3 E 1 4n 2 | f | 2 n 2 g 2 )
x k =2 a 3 cos( ϕk 2π 3 ),k=0,1,2
ϕ= 1 3 cos 1 ( 3b 2a 3 a ).
E D1 =2 a 3 cos( ϕ 4π 3 )+ 1 3 ( E 1 + E 2 + E 3 )
E D2 =2 a 3 cos( ϕ 2π 3 )+ 1 3 ( E 1 + E 2 + E 3 )
E D3 =2 a 3 cos( ϕ )+ 1 3 ( E 1 + E 2 + E 3 )
| D i (n) = a i |1+ b i |2+ c i |3,    ( i=1,2,3 )
1= | a i | 2 + | b i | 2 + | c i | 2 .
| D i (n) = 1 Ξ i [ ( ( E Di E 2 ) g n 4 | f | 2 g( E Di E 3 ) )|1+(1)|2+( 2 | f | 2 ( E Di E 3 ) )|3 ],
Ξ i = ( E Di E 2 g n 4 | f | 2 g( E Di E 3 ) ) 2 + 1 2 + ( 2 | f | 2 ( E Di E 3 ) ) 2 .
m ^ =Tr( μ (m) , ρ ˜ ) =( μ 21 (m) ρ ˜ 12 + μ 31 (m) ρ ˜ 13 + μ 32 (m) ρ ˜ 23 )+c.c.
m ^ (2) = [ μ 13 me ρ ˜ 31 (2) ] y +c.c.
ρ ˜ 13 (2) = ρ ˜ 12 (1) V ˜ 23 (1) Δ 13 +i Γ 13 ,
m ^ (2) = { a i b i * D i | μ 12 (e) ξ μ eff (m) L ^ ' O ^ ' + σ ^ + | D i +h.c. } y = a i b i * n| 11 | 21,1 | c i * μ eff (m) L ^ ' O ^ ' + b i | 210 | 10 |n+h.c. =2 a i b i * b i c i * μ eff (m) 11 | 21,1 | 21,1 | 11 +c.c. =2c μ 12 (e) ( a i b i * b i c i * +c.c.)
m ^ (3) = μ 23 (m) ρ 32 (3) +c.c.
ρ ˜ 23 (3) = V 21 (1) ρ ˜ 13 (2) Δ 23 +i Γ 23 ,
m ^ (3) = { a i c i * D i | μ 21 (e) ξ2 μ eff (m) σ ^ + | D i +h.c. } y =2 a i c i * n| 10 | 210 | b i * μ eff (m) σ ^ + a i | 100 | 00 |n+h.c. =2 a i * b i a i c i * μ eff (m) 10 | 210 | 210 | 10 +c.c. =2c μ 12 (e) ( a i * b i a i c i * +c.c.)
m ^ (ω) =2c μ 12 (e) { j=1 3 ( a j b j * a j c j * +c.c.+ a j b j * b j c j * +c.c. ) 2 } 1/2 .
H ^ int (m) =f L ^ ' O ^ + ' a ^ + +h.c.
[ H ^ int (m) , J ^ ]=0
[ H ^ int (m) , J ^ z ]=0.
l'O'j'm' |[ H ^ int (m) , J ^ z ]| lOjm =(mm') l'O'j'm' | H ^ int (m) | lOjm =0.
m l + m o =m ' l +m ' o .
H= 1 2 I ω 2 = 1 2 ω ¯ O ¯ ,
O ¯ = 0 Δt ( d O ¯ dt ) dt.
d J ¯ /dt=d( L ¯ + O ¯ )/dt=0,
O ¯ = 0 Δt ( d L ¯ dt ) dt.
O ¯ = 0 Δt ( m ¯ × B ¯ )dt .
L ¯ = m e r ¯ × d r ¯ dt ,
L= m e r 2 ω 0 ,
O ¯ m e h 2 ω φ r ^ 0 = m e hr ω 0 r ^ 0 ,
O= m e hr ω 0 .
ηO/Lh/r.
m ¯ =η( e L ¯ 2 m e )
O ¯ = 0 Δt ( eη 2 m e ) ( L ¯ × B ¯ )dt.
O ¯ =( eη 2 m e )( 2 L ¯ 0 × B ¯ 0 * 4 )Δt.
H= 1 2I eη 2 m e O ¯ ( L ¯ 0 × B ¯ 0 * 2 )Δt= 1 2I μ ¯ O ^ ( L ¯ 0 × B ¯ 0 * 2 )Δt,
Δ L ¯ =( e 2 m e )( L ¯ 0 × B ¯ 0 * 2 )Δt= O ^ .
Δt= 4 m e e B 0 * = 4 ω c ,
H int =( 2 ω φ ω c ) μ ¯ O ^ (L'× B ¯ )= μ eff O ¯ '(L'× B ¯ ).
μ eff =( 2 ω 0 ω c ) μ 0 (m) .
L ¯ '× y ^ = L x ' z ^ L z ' x ^
O ¯ '( L ¯ '×B y ^ )= O ¯ ' ( L x ' z ^ L z ' x ^ )B=( O z ' L x ' O x ' L z ' )B
H ^ int = μ eff O ^ x ' L ^ z ' B ^ ,
L ^ z ' = 1 2i ( L ^ + ' L ^ ' )
O ^ x ' = 1 2i ( O ^ + ' O ^ ' )
H ^ int = μ eff 1 2i ( O ^ + ' O ^ ' ) 1 2i ( L ^ + ' L ^ ' )i( a ^ + a ^ )ξ/c,
H ^ int =i μ eff ( O ^ + ' L ^ ' a ^ + O ^ ' L ^ + ' a ^ )ξ/c
H ^ int =f L ^ ' O ^ + ' a ^ + +h.c.,

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