Abstract

The state mixings necessary to mediate three new optical nonlinearities are shown to arise simultaneously and automatically in a 2-level atom with an = 0 ground state and an = 1 excited state that undergoes a sequence of electric and magnetic dipole-allowed transitions. The treatment is based on an extension of dressed state theory that includes quantized electric and magnetic field interactions. Magneto-electric rectification, transverse magnetization, and second-harmonic generation are shown to constitute a family of nonlinear effects that can take place regardless of whether inversion is a symmetry of the initial unperturbed system or not. Interactions driven jointly by the optical electric and magnetic fields produce dynamic symmetry-breaking that accounts for the frequency, the intensity dependence, and the polarization of induced magnetization in prior experiments. This strong field quantum model explains not only how a driven 2-level system may develop nonlinear dipole moments that are forbidden between or within its stationary states, but it also broadens the class of materials suitable for optical energy conversion applications and magnetic field generation with light so as to include all transparent dielectrics.

© 2014 Optical Society of America

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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, 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, 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 centers in diamond,” Nat. Phys. 7, 789–793 (2011).
    [CrossRef]
  4. C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and Th. Rasing, “All-optical magnetic recording with circularly polarized pulses,” Phys. Rev. Lett. 99, 047601 (2007).
    [CrossRef]
  5. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984), pp. 268–269.
  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, 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, 1106–1117 (2008).
    [CrossRef]
  8. W. M. Fisher and S. C. Rand, “Dependence of optical magnetic response on molecular electronic structure,” J. Lumin. 129, 1407–1409 (2009).
    [CrossRef]
  9. W. M. Fisher and S. C. Rand, “Optically-induced charge separation and terahertz emission in unbiased dielectrics,” J. Appl. Phys. 109, 064903 (2011).
    [CrossRef]
  10. W. M. Fisher and S. C. Rand, “Light-induced dynamics in the Lorentz oscillator model with magnetic forces,” Phys. Rev. A 82, 013802 (2010).
    [CrossRef]
  11. S. C. Rand, “Quantum theory of coherent transverse optical magnetism,” J. Opt. Soc. Am. B 26, B120–B129 (2009); Erratum J. Opt. Soc. Am. B 27, 1983 (2010); Erratum J. Opt. Soc. Am. B 28, 1792 (2011).
    [CrossRef]
  12. P. S. Pershan, “Nonlinear optical properties of solids: Energy considerations,” Phys. Rev. 130, 919–929 (1963).
    [CrossRef]
  13. M. Frasca, “A modern review of the two-level approximation,” Annals of Physics 306, 193–208 (2003).
    [CrossRef]
  14. C. Cohen-Tannoudji, Atom-photon Interactions: Basic Processes and Applications (J. Wiley & Sons Inc., 1992), pp. 411–417.
  15. M. Abramowitz and I. E. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1972 Appl. Math. Series 55, Tenth printing), pp. 17.
  16. K. Konishi and G. Paffuti, Quantum Mechanics: A New Introduction (Oxford University, 2009), pp. 116–117.

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, 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, 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 centers in diamond,” Nat. Phys. 7, 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, 064903 (2011).
[CrossRef]

2010 (1)

W. M. Fisher and S. C. Rand, “Light-induced dynamics in the Lorentz oscillator model with magnetic forces,” Phys. Rev. A 82, 013802 (2010).
[CrossRef]

2009 (2)

2008 (1)

2007 (2)

C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and Th. Rasing, “All-optical magnetic recording with circularly polarized pulses,” Phys. Rev. Lett. 99, 047601 (2007).
[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, 093901 (2007).
[CrossRef] [PubMed]

2003 (1)

M. Frasca, “A modern review of the two-level approximation,” Annals of Physics 306, 193–208 (2003).
[CrossRef]

1963 (1)

P. S. Pershan, “Nonlinear optical properties of solids: Energy considerations,” Phys. Rev. 130, 919–929 (1963).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. E. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1972 Appl. Math. Series 55, Tenth printing), pp. 17.

Awschalom, D. D.

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

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, 84–87 (2011).
[CrossRef] [PubMed]

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, 097402 (2012).
[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, 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, 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 centers in diamond,” Nat. Phys. 7, 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, 84–87 (2011).
[CrossRef] [PubMed]

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, 097402 (2012).
[CrossRef] [PubMed]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, Atom-photon Interactions: Basic Processes and Applications (J. Wiley & Sons Inc., 1992), pp. 411–417.

Fisher, W. M.

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

W. M. Fisher and S. C. Rand, “Light-induced dynamics in the Lorentz oscillator model with magnetic forces,” Phys. Rev. A 82, 013802 (2010).
[CrossRef]

W. M. Fisher and S. C. Rand, “Dependence of optical magnetic response on molecular electronic structure,” J. Lumin. 129, 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, 1106–1117 (2008).
[CrossRef]

Frasca, M.

M. Frasca, “A modern review of the two-level approximation,” Annals of Physics 306, 193–208 (2003).
[CrossRef]

Fuchs, G. D.

G. D. Fuchs, G. Burkard, P. V. Klimov, and D. D. Awschalom, “A quantum memory intrinsic to single nitrogen-vacancy centers in diamond,” Nat. Phys. 7, 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, 097402 (2012).
[CrossRef] [PubMed]

Hansteen, F.

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

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, 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, 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, 097402 (2012).
[CrossRef] [PubMed]

Itoh, A.

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

Kimel, A. V.

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

Kirilyuk, A.

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

Klimov, P. V.

G. D. Fuchs, G. Burkard, P. V. Klimov, and D. D. Awschalom, “A quantum memory intrinsic to single nitrogen-vacancy centers in diamond,” Nat. Phys. 7, 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, 84–87 (2011).
[CrossRef] [PubMed]

Konishi, K.

K. Konishi and G. Paffuti, Quantum Mechanics: A New Introduction (Oxford University, 2009), pp. 116–117.

Landau, L. D.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984), pp. 268–269.

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984), pp. 268–269.

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, 1106–1117 (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, 093901 (2007).
[CrossRef] [PubMed]

Paffuti, G.

K. Konishi and G. Paffuti, Quantum Mechanics: A New Introduction (Oxford University, 2009), pp. 116–117.

Pershan, P. S.

P. S. Pershan, “Nonlinear optical properties of solids: Energy considerations,” Phys. Rev. 130, 919–929 (1963).
[CrossRef]

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, 097402 (2012).
[CrossRef] [PubMed]

Pitaevskii, L. P.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984), pp. 268–269.

Rand, S. C.

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

W. M. Fisher and S. C. Rand, “Light-induced dynamics in the Lorentz oscillator model with magnetic forces,” Phys. Rev. A 82, 013802 (2010).
[CrossRef]

W. M. Fisher and S. C. Rand, “Dependence of optical magnetic response on molecular electronic structure,” J. Lumin. 129, 1407–1409 (2009).
[CrossRef]

S. C. Rand, “Quantum theory of coherent transverse optical magnetism,” J. Opt. Soc. Am. B 26, B120–B129 (2009); Erratum J. Opt. Soc. Am. B 27, 1983 (2010); Erratum J. Opt. Soc. Am. B 28, 1792 (2011).
[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, 1106–1117 (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, 093901 (2007).
[CrossRef] [PubMed]

Rasing, Th.

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

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, 097402 (2012).
[CrossRef] [PubMed]

Stanciu, C. D.

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

Stegun, I. E.

M. Abramowitz and I. E. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1972 Appl. Math. Series 55, Tenth printing), pp. 17.

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, 097402 (2012).
[CrossRef] [PubMed]

Tsukamoto, A.

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

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, 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, 097402 (2012).
[CrossRef] [PubMed]

Annals of Physics (1)

M. Frasca, “A modern review of the two-level approximation,” Annals of Physics 306, 193–208 (2003).
[CrossRef]

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, 064903 (2011).
[CrossRef]

J. Lumin. (1)

W. M. Fisher and S. C. Rand, “Dependence of optical magnetic response on molecular electronic structure,” J. Lumin. 129, 1407–1409 (2009).
[CrossRef]

J. Opt. Soc. Am. B (2)

Nat. Phys. (1)

G. D. Fuchs, G. Burkard, P. V. Klimov, and D. D. Awschalom, “A quantum memory intrinsic to single nitrogen-vacancy centers in diamond,” Nat. Phys. 7, 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, 84–87 (2011).
[CrossRef] [PubMed]

Phys. Rev. (1)

P. S. Pershan, “Nonlinear optical properties of solids: Energy considerations,” Phys. Rev. 130, 919–929 (1963).
[CrossRef]

Phys. Rev. A (1)

W. M. Fisher and S. C. Rand, “Light-induced dynamics in the Lorentz oscillator model with magnetic forces,” Phys. Rev. A 82, 013802 (2010).
[CrossRef]

Phys. Rev. Lett. (3)

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, 093901 (2007).
[CrossRef] [PubMed]

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, 097402 (2012).
[CrossRef] [PubMed]

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

Other (4)

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984), pp. 268–269.

C. Cohen-Tannoudji, Atom-photon Interactions: Basic Processes and Applications (J. Wiley & Sons Inc., 1992), pp. 411–417.

M. Abramowitz and I. E. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, 1972 Appl. Math. Series 55, Tenth printing), pp. 17.

K. Konishi and G. Paffuti, Quantum Mechanics: A New Introduction (Oxford University, 2009), pp. 116–117.

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Figures (6)

Fig. 1:
Fig. 1:

Ground and excited energy states of a 2-level atom having an allowed ED transition at frequency ω0. The ground and excited eigenstates are assumed to be = 0 and = 1, respectively.

Fig. 2:
Fig. 2:

(a) Dressed state picture of the energy levels driven by light of frequency ω at a detuning from resonance of Δ21. (b) Illustration of the explicit admixtures of the doubly-dressed states mediated by electric and magnetic dipole interactions, together with the linear and second-order dipole moments that arise between them. Solid arrows are ED-allowed transitions at the indicated frequencies. Dashed arrows are MD-allowed transitions at the optical frequency. For simplicity, only transitions between dressed states |Di〉 with i = 1 are shown.

Fig. 3:
Fig. 3:

Log-log plots of the absolute magnitudes of linear and nonlinear moments versus photon number in the doubly-dressed state picture using the four-state basis. Each plot shows five moments coded by symbols in the legend: p i i ( 1 ) ( ω ) , p i i ( 2 ) ( 0 ) , p i i ( 2 ) ( 2 ω ) , m i i ( 2 ) ( ω ) l / c, and m i i ( 2 ) ( ω ) u / c. The magnetic moments were divided by a factor of the speed of light, c, to make the units for all dipole moments C·m. Interaction strengths were g = 10 ω s 1 / 2 and f = g/204. Detuning was fixed at Δ21/ω0 = 0.1.

Fig. 4:
Fig. 4:

Frequency dependence of the absolute magnitude of dipole moments between various doubly-dressed energy states in the four-state basis. (a) p ^ i i ( 1 ) ( ω ) , (b) p ^ i i ( 2 ) ( 0 ) , (c) p ^ i i ( 2 ) ( 2 ω ) , (d) m ^ i i ( 2 ) ( ω ) l / c, and (e) m ^ i i ( 2 ) ( ω ) u / c between dressed states with indices i = 1 (solid), and i = 2 (dashed). (f) Expansion of the region in plot (e) enclosed by a rectangle to show the (small) 2-photon resonance at Δ31 = 0 for the magnetization of the upper transition. Each curve is dotted outside of the range [0.85ω0, 1.15ω0] to show reasonable values for which the RWA is valid for the electric interaction. All plots are for n = 109 with interaction strengths of g = 10 ω s 1 / 2 and f = g/204.

Fig. 5:
Fig. 5:

Plot of enhancements for dipole moments between dressed states with index i = 1 near two-photon resonance (Δ31/ω0 = 2 × 10−6). Interaction strengths were taken to be g = 10 ω s 1 / 2 and f = g/204. Note in particular that p 11 ( 2 ) ( 2 ω ) and m 11 ( 4 ) ( ω ) u / c increase over the results shown in Fig. 3(a).

Fig. 6:
Fig. 6:

Log-log plots of the absolute magnitudes of the corrections to nonlinear moments in the expanded eight-state basis. Each plot shows a selection of moments coded by symbols in the legend for a different dressed state index. Plots (a) and (b) show p i i ( 2 ) ( 0 ) , p i i ( 2 ) ( 2 ω ) , m i i ( 2 ) ( ω ) l / c, m i i ( 2 ) ( ω ) u / c, and m i i ( 2 ) ( ω ) 1 / 2 / c for indices i = 1 and i = 2, respectively. Plots (c) and (d) are for indices i = 7 and i = 8, respectively, and only a selection of the nonlinear effects appear for the ranges chosen. Interaction strengths were g = 10 ω s 1 / 2 and f = g/204, and the detuning was fixed at Δ21/ω0 = 0.1.

Equations (36)

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

| 1 | 1 , 0 , 0 | n ,
| 2 | 2 , 1 , 0 | n 1 ,
| 3 | 2 , 1 , 1 | n 2 ,
| 4 | 2 , 1 , 1 | n .
H ^ a f = H ^ atom + H ^ field = ω 0 2 σ ^ z + ω a ^ + a ^ ,
E 1 = ω 0 2 + n ω ,
E 2 = E 1 + Δ 21 = ω 0 2 + ( n 1 ) ω ,
E 3 = E 2 ω = ω 0 2 + ( n 2 ) ω ,
E 4 = E 2 + ω = ω 0 2 + n ω .
H ^ int = H ^ int ( e ) + H ^ int ( m ) = g ( σ ^ + a ^ + h . c . ) + f ( L ^ + a ^ + h . c . + L ^ + a ^ + + h . c . ) .
H ^ = H ^ atom + H ^ field + H ^ int .
H ^ | D i ( n ) = E D i | D i ( n ) ,
( E 4 0 f 2 n 0 0 E 3 f 2 ( n 1 ) 0 f 2 n f 2 ( n 1 ) E 2 g n 0 0 g n E 1 ) ( d i c i b i a i ) = E D i ( d i c i b i a i ) ,
| D i ( n ) = a i | 1 , 0 , 0 | n + b i | 2 , 1 , 0 | n 1 + c i | 2 , 1 , 1 | n 2 + d i | 2 , 1 , 1 | n ,
| 5 | 2 , 1 , 1 | n ,
| 6 | 2 , 1 , 1 | n 2 ,
| 7 | 2 , 1 , 0 | n + 1 ,
| 8 | 2 , 1 , 0 | n 3 .
E 5 = ω 0 2 + n ω ,
E 6 = ω 0 2 + ( n 2 ) ω ,
E 7 = ω 0 2 + ( n + 1 ) ω ,
E 8 = ω 0 2 + ( n 3 ) ω .
( E 8 0 t 0 0 t 0 0 0 E 7 0 s s 0 0 0 t 0 E 6 0 0 0 p 0 0 s 0 E 5 0 0 q 0 0 s 0 0 E 4 0 q 0 t 0 0 0 0 E 3 p 0 0 0 p q q p E 2 r 0 0 0 0 0 0 r E 1 ) ( h i g i f i e i d i c i b i a i ) = E D i ( h i g i f i e i d i c i b i a i ) ,
| D i ( n ) = a i | 1 , 0 , 0 | n + b i | 2 , 1 , 0 | n 1 + c i | 2 , 1 , 1 | n 2 + d i | 2 , 1 , 1 | n + e i | 2 , 1 , 1 | n + f i | 2 , 1 , 1 | n 2 + g i | 2 , 1 , 0 | n + 1 + h i | 2 , 1 , 0 | n 3
| a i | 2 + | b i | 2 + | c i | 2 + | d i | 2 + | e i | 2 + | f i | 2 + | g i | 2 + | h i | 2 = 1 ,
p ^ 11 ( 1 ) ( ω ) = D 1 ( n + 1 ) | e r ^ | D 1 ( n ) + h . c . = a 1 b 1 * n | 2 , 1 , 0 | e r ^ | 1 , 0 , 0 | n + h . c . = μ ( e ) ( a 1 b 1 * + a 1 * b 1 ) x ^ .
p ^ 11 ( 2 ) ( 0 ) = D 1 ( n ) | e r ^ | D 1 ( n ) + h . c . = ( a 1 d 1 * n | 2 , 1 , 1 | e r ^ | 1 , 0 , 0 | n + a 1 * d 1 n | 1 , 0 , 0 | e r ^ | 2 , 1 , 1 | n ) + h . c . = μ ( e ) 2 { [ a 1 d 1 * ( z ^ + i y ^ ) + a 1 * d 1 ( z ^ i y ^ ) ] + h . c . } = μ ( e ) 2 2 ( a 1 d 1 * + a 1 * d 1 ) z ^ .
p ^ 11 ( 2 ) ( 2 ω ) = μ ( e ) 1 2 ( a 1 c 1 * + a 1 * c 1 ) z ^ ,
m ^ 11 ( 2 ) ( ω ) l = μ ( m ) ( a 1 b 1 * b 1 d 1 * + a 1 * b 1 b 1 * d 1 ) y ^ ,
m ^ 11 ( 2 ) ( ω ) u = μ ( m ) ( a 1 b 1 * b 1 c 1 * + a 1 * b 1 b 1 * c 1 ) y ^ .
p ^ 11 ( 2 ) ( 0 ) = μ ( e ) 2 2 ( a 1 e 1 * + a 1 * e 1 ) z ^ ,
p ^ 11 ( 2 ) ( 2 ω ) = μ ( e ) 1 2 ( a 1 f 1 * + a 1 * f 1 ) z ^ ,
m ^ 11 ( 2 ) ( ω ) l = μ ( m ) ( a 1 b 1 * b 1 e 1 * + a 1 * b 1 b 1 * e 1 ) y ^ ,
m ^ 11 ( 2 ) ( ω ) u = μ ( m ) ( a 1 b 1 * b 1 f 1 * + a 1 * b 1 b 1 * f 1 ) y ^ .
m ^ 11 ( 4 ) ( ω ) 1 = μ ( m ) ( a 1 g 1 * g 1 d 1 * + a 1 * g 1 g 1 * d 1 ) y ^ ,
m ^ 11 ( 4 ) ( ω ) 2 = μ ( m ) ( a 1 g 1 * g 1 e 1 * + a 1 * g 1 g 1 * e 1 ) y ^ .

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