Abstract

Spin noise spectroscopy (SNS) is a new method for studying magnetic resonance and spin dynamics that has gained, in the last several years, a considerable popularity. The method is based on measuring magnetization noise of a paramagnet using the Faraday rotation technique. In strong contrast with methods of nonlinear optics, the spectroscopy of spin noise is considered to be essentially nonperturbative. At the same time, presently, it became clear that the SNS, as an optical technique, demonstrates abilities lying far beyond the bounds of conventional linear optics. Specifically, the SNS allows one to penetrate inside an inhomogeneously broadened absorption band and to determine its homogeneous width, to realize a sort of pump-probe spectroscopy without any optical nonlinearity, to probe a bulk inhomogeneous medium by focal point of a probe beam, etc. This may seem especially puzzling when taken into account that SNS can be considered just as a version of Raman spectroscopy, which is known to be deprived of such abilities. Understanding of these paradoxical features of SNS technique is required for the present-day applications of SNS and its further development. In this paper, we present a general analysis of this apparent inconsistency from the viewpoint of distinction between spectroscopy of the light intensity and of the light field and provide its resolution.

© 2015 Optical Society of America

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References

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  39. The electric field at the detector is, as a rule, inhomogeneous due to, e.g., inhomogeneity of the sample, effects of diffraction, etc. The way how the signal is picked up and/or averaged in the particular setup may change the results quantitatively.
  40. In the latter case the field propagating backwards contains the contribution ∝ E+ due to specular nonresonant reflection from the sample surface.
  41. There is also an interference of P+0 and P+′, which is briefly discussed below.
  42. There could be minor contributions if spins can freely propagate and go between the spots illuminated by the first and the second beams, cf. [24].
  43. In inhomogeneous systems Eq. (18) should be modified following approaches of [16,18,19] to take the inhomogeneity into account.

2014 (3)

L. Yang, P. Glasenapp, A. Greilich, D. Reuter, A. D. Wieck, D. R. Yakovlev, M. Bayer, and S. A. Crooker, “Two-colour spin noise spectroscopy and fluctuation correlations reveal homogeneous linewidths within quantum-dot ensembles,” Nat. Commun. 5, 4949 (2014).
[Crossref] [PubMed]

G. G. Kozlov and V. S. Zapasskii, “Intensity-related optical susceptibility and ‘active’ noise spectroscopy,” J. of Phys. B 47, 185401 (2014).
[Crossref]

J. Hübner, F. Berski, R. Dahbashi, and M. Oestreich, “The rise of spin noise spectroscopy in semiconductors: from acoustic to GHz frequencies,” Phys. Stat. Solidi B 251, 1824 (2014).
[Crossref]

2013 (3)

V. S. Zapasskii, “Spin-noise spectroscopy: from proof of principle to applications,” Adv. Opt. Photon. 5, 131 (2013).
[Crossref]

Yu. V. Pershin, V. A. Slipko, D. Roy, and N. A. Sinitsyn, “Two-beam spin noise spectroscopy,” Appl. Phys. Lett. 102, 202405 (2013).
[Crossref]

V. S. Zapasskii, A. Greilich, S. A. Crooker, Y. Li, G. G. Kozlov, D. R. Yakovlev, D. Reuter, A. D. Wieck, and M. Bayer, “Optical spectroscopy of spin noise,” Phys. Rev. Lett. 110, 176601 (2013).
[Crossref] [PubMed]

2012 (3)

M. M. Glazov, “Coherent spin dynamics of electrons and excitons in nanostructures (a review),” Phys. Solid State 54, 1 (2012).
[Crossref]

A. Perrin, R. Bücker, S. Manz, T. Betz, C. Koller, T. Plisson, T. Schumm, and J. Schmiedmayer, “Hanbury Brown and Twiss correlations across the BoseEinstein condensation threshold,” Nat. Phys. 8, 195 (2012).
[Crossref]

M. M. Glazov and E. L. Ivchenko, “Spin noise in quantum dot ensembles,” Phys. Rev. B 86, 115308 (2012).
[Crossref]

2011 (1)

Y Lahini, Y Bromberg, Y. Shechtman, A. Szameit, D. N. Christodoulides, R. Morandotti, and Y Silberberg, “Hanbury-Brown and Twiss correlations of Anderson localized waves,” Phys. Rev. A 84, 041806 (2011).
[Crossref]

2010 (2)

G. M. Müller, M. Oestreich, M. Römer, and J. Hübner, “Semiconductor spin noise spectroscopy: fundamentals, accomplishments, and challenges,” Physica E 43, 569 (2010).
[Crossref]

Y Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury-Brown and Twiss interferometry with interacting photons,” Nat. Photonics 4, 721 (2010).
[Crossref]

2009 (1)

M. Römer, J. Hübner, and M. Oestreich, “Spatially resolved doping concentration measurement in semiconductors via spin noise spectroscopy,” Appl. Phys. Lett. 94, 112105 (2009).
[Crossref]

2008 (1)

V. S. Zapasskii and G. G. Kozlov, “Slow light and slow current,” Opt. Spectrosc. 104, 95 (2008).
[Crossref]

2007 (1)

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury BrownTwiss effect for bosons and fermions,” Nature 445, 402 (2007).
[Crossref] [PubMed]

2004 (1)

S. A. Crooker, D. G. Rickel, A. V. Balatsky, and D. L. Smith, “Spectroscopy of spontaneous spin noise as a probe of spin dynamics and magnetic resonance,” Nature 431, 49 (2004).
[Crossref] [PubMed]

2001 (1)

M. Dantus, “Coherent Nonlinear Spectroscopy: From Femtosecond Dynamics to Control,” Annu. Rev. Phys. Chem. 52, 639 (2001).
[Crossref] [PubMed]

1994 (1)

J. Erland, K.-H. Pantke, V. Mizeikis, V. G. Lyssenkot, and J. M. Hvam, “Spectrally resolved four-wave mixing in semiconductors: influence of inhomogeneous broadening,” Phys. Rev. B 5015047 (1994).
[Crossref]

1990 (1)

E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[Crossref]

1989 (1)

1983 (2)

B. M. Gorbovitskii and V. I. Perel, “Aleksandrov and Zapasskii experiment and the Raman effect,” Opt. Spectrosc. 54, 229 (1983).

E. B. Aleksandrov, Yu. M. Golubev, A. V. Lomakin, and V. A. Noskin, “Intensity-fluctuation spectroscopy of optical fields with non-Gaussian statistics,” Sov. Phys. Usp. 26, 643 (1983).
[Crossref]

1981 (1)

E. Aleksandrov and V. Zapasskii, “Magnetic resonance in the Faraday-rotation noise spectrum,” J. Experim. Theor. Phys. 54, 64 (1981).

1957 (3)

R.M. Sillitto, “Correlation between events in photon detectors,” Nature 179, 1127 (1957).
[Crossref]

P. Fellgett, “Question of correlation between photons in coherent beams of light,” Nature 179, 956 (1957).
[Crossref]

G.A. Rebka and R.V. Pound, “Time-correlated photons,” Nature 180, 1035 (1957).
[Crossref]

1956 (1)

R. Hanbury-Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27 (1956).
[Crossref]

1955 (1)

A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691 (1955).
[Crossref]

Aleksandrov, E.

E. Aleksandrov and V. Zapasskii, “Magnetic resonance in the Faraday-rotation noise spectrum,” J. Experim. Theor. Phys. 54, 64 (1981).

Aleksandrov, E. B.

E. B. Aleksandrov, Yu. M. Golubev, A. V. Lomakin, and V. A. Noskin, “Intensity-fluctuation spectroscopy of optical fields with non-Gaussian statistics,” Sov. Phys. Usp. 26, 643 (1983).
[Crossref]

Aspect, A.

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury BrownTwiss effect for bosons and fermions,” Nature 445, 402 (2007).
[Crossref] [PubMed]

Balatsky, A. V.

S. A. Crooker, D. G. Rickel, A. V. Balatsky, and D. L. Smith, “Spectroscopy of spontaneous spin noise as a probe of spin dynamics and magnetic resonance,” Nature 431, 49 (2004).
[Crossref] [PubMed]

Bayer, M.

L. Yang, P. Glasenapp, A. Greilich, D. Reuter, A. D. Wieck, D. R. Yakovlev, M. Bayer, and S. A. Crooker, “Two-colour spin noise spectroscopy and fluctuation correlations reveal homogeneous linewidths within quantum-dot ensembles,” Nat. Commun. 5, 4949 (2014).
[Crossref] [PubMed]

V. S. Zapasskii, A. Greilich, S. A. Crooker, Y. Li, G. G. Kozlov, D. R. Yakovlev, D. Reuter, A. D. Wieck, and M. Bayer, “Optical spectroscopy of spin noise,” Phys. Rev. Lett. 110, 176601 (2013).
[Crossref] [PubMed]

Berski, F.

J. Hübner, F. Berski, R. Dahbashi, and M. Oestreich, “The rise of spin noise spectroscopy in semiconductors: from acoustic to GHz frequencies,” Phys. Stat. Solidi B 251, 1824 (2014).
[Crossref]

Betz, T.

A. Perrin, R. Bücker, S. Manz, T. Betz, C. Koller, T. Plisson, T. Schumm, and J. Schmiedmayer, “Hanbury Brown and Twiss correlations across the BoseEinstein condensation threshold,” Nat. Phys. 8, 195 (2012).
[Crossref]

Bloembergen, N.

N. Bloembergen, Nonlinear Optics (World Scientific, 1996).

Boiron, D.

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury BrownTwiss effect for bosons and fermions,” Nature 445, 402 (2007).
[Crossref] [PubMed]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic press, 2003).

Bromberg, Y

Y Lahini, Y Bromberg, Y. Shechtman, A. Szameit, D. N. Christodoulides, R. Morandotti, and Y Silberberg, “Hanbury-Brown and Twiss correlations of Anderson localized waves,” Phys. Rev. A 84, 041806 (2011).
[Crossref]

Y Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury-Brown and Twiss interferometry with interacting photons,” Nat. Photonics 4, 721 (2010).
[Crossref]

Bücker, R.

A. Perrin, R. Bücker, S. Manz, T. Betz, C. Koller, T. Plisson, T. Schumm, and J. Schmiedmayer, “Hanbury Brown and Twiss correlations across the BoseEinstein condensation threshold,” Nat. Phys. 8, 195 (2012).
[Crossref]

Carmichael, H.

H. Carmichael, An Open System Approach to Quantum Optics (Springer-Verlag, 1993).

Chang, H.

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury BrownTwiss effect for bosons and fermions,” Nature 445, 402 (2007).
[Crossref] [PubMed]

Christodoulides, D. N.

Y Lahini, Y Bromberg, Y. Shechtman, A. Szameit, D. N. Christodoulides, R. Morandotti, and Y Silberberg, “Hanbury-Brown and Twiss correlations of Anderson localized waves,” Phys. Rev. A 84, 041806 (2011).
[Crossref]

Crooker, S. A.

L. Yang, P. Glasenapp, A. Greilich, D. Reuter, A. D. Wieck, D. R. Yakovlev, M. Bayer, and S. A. Crooker, “Two-colour spin noise spectroscopy and fluctuation correlations reveal homogeneous linewidths within quantum-dot ensembles,” Nat. Commun. 5, 4949 (2014).
[Crossref] [PubMed]

V. S. Zapasskii, A. Greilich, S. A. Crooker, Y. Li, G. G. Kozlov, D. R. Yakovlev, D. Reuter, A. D. Wieck, and M. Bayer, “Optical spectroscopy of spin noise,” Phys. Rev. Lett. 110, 176601 (2013).
[Crossref] [PubMed]

S. A. Crooker, D. G. Rickel, A. V. Balatsky, and D. L. Smith, “Spectroscopy of spontaneous spin noise as a probe of spin dynamics and magnetic resonance,” Nature 431, 49 (2004).
[Crossref] [PubMed]

D. Roy, L. Yang, S. A. Crooker, and N. A. Sinitsyn, “Cross-correlation spin noise spectroscopy of heterogeneous interacting spin systems,” arXiv:1408.5399v1 (2014).

Dahbashi, R.

J. Hübner, F. Berski, R. Dahbashi, and M. Oestreich, “The rise of spin noise spectroscopy in semiconductors: from acoustic to GHz frequencies,” Phys. Stat. Solidi B 251, 1824 (2014).
[Crossref]

Dantus, M.

M. Dantus, “Coherent Nonlinear Spectroscopy: From Femtosecond Dynamics to Control,” Annu. Rev. Phys. Chem. 52, 639 (2001).
[Crossref] [PubMed]

Erland, J.

J. Erland, K.-H. Pantke, V. Mizeikis, V. G. Lyssenkot, and J. M. Hvam, “Spectrally resolved four-wave mixing in semiconductors: influence of inhomogeneous broadening,” Phys. Rev. B 5015047 (1994).
[Crossref]

Fellgett, P.

P. Fellgett, “Question of correlation between photons in coherent beams of light,” Nature 179, 956 (1957).
[Crossref]

Forrester, A. T.

A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691 (1955).
[Crossref]

Glasenapp, P.

L. Yang, P. Glasenapp, A. Greilich, D. Reuter, A. D. Wieck, D. R. Yakovlev, M. Bayer, and S. A. Crooker, “Two-colour spin noise spectroscopy and fluctuation correlations reveal homogeneous linewidths within quantum-dot ensembles,” Nat. Commun. 5, 4949 (2014).
[Crossref] [PubMed]

Glazov, M. M.

M. M. Glazov and E. L. Ivchenko, “Spin noise in quantum dot ensembles,” Phys. Rev. B 86, 115308 (2012).
[Crossref]

M. M. Glazov, “Coherent spin dynamics of electrons and excitons in nanostructures (a review),” Phys. Solid State 54, 1 (2012).
[Crossref]

Golubev, Yu. M.

E. B. Aleksandrov, Yu. M. Golubev, A. V. Lomakin, and V. A. Noskin, “Intensity-fluctuation spectroscopy of optical fields with non-Gaussian statistics,” Sov. Phys. Usp. 26, 643 (1983).
[Crossref]

Gorbovitskii, B. M.

B. M. Gorbovitskii and V. I. Perel, “Aleksandrov and Zapasskii experiment and the Raman effect,” Opt. Spectrosc. 54, 229 (1983).

Greilich, A.

L. Yang, P. Glasenapp, A. Greilich, D. Reuter, A. D. Wieck, D. R. Yakovlev, M. Bayer, and S. A. Crooker, “Two-colour spin noise spectroscopy and fluctuation correlations reveal homogeneous linewidths within quantum-dot ensembles,” Nat. Commun. 5, 4949 (2014).
[Crossref] [PubMed]

V. S. Zapasskii, A. Greilich, S. A. Crooker, Y. Li, G. G. Kozlov, D. R. Yakovlev, D. Reuter, A. D. Wieck, and M. Bayer, “Optical spectroscopy of spin noise,” Phys. Rev. Lett. 110, 176601 (2013).
[Crossref] [PubMed]

Gudmundsen, R. A.

A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691 (1955).
[Crossref]

Hanbury-Brown, R.

R. Hanbury-Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27 (1956).
[Crossref]

Hogervorst, W.

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury BrownTwiss effect for bosons and fermions,” Nature 445, 402 (2007).
[Crossref] [PubMed]

Hübner, J.

J. Hübner, F. Berski, R. Dahbashi, and M. Oestreich, “The rise of spin noise spectroscopy in semiconductors: from acoustic to GHz frequencies,” Phys. Stat. Solidi B 251, 1824 (2014).
[Crossref]

G. M. Müller, M. Oestreich, M. Römer, and J. Hübner, “Semiconductor spin noise spectroscopy: fundamentals, accomplishments, and challenges,” Physica E 43, 569 (2010).
[Crossref]

M. Römer, J. Hübner, and M. Oestreich, “Spatially resolved doping concentration measurement in semiconductors via spin noise spectroscopy,” Appl. Phys. Lett. 94, 112105 (2009).
[Crossref]

Hvam, J. M.

J. Erland, K.-H. Pantke, V. Mizeikis, V. G. Lyssenkot, and J. M. Hvam, “Spectrally resolved four-wave mixing in semiconductors: influence of inhomogeneous broadening,” Phys. Rev. B 5015047 (1994).
[Crossref]

Ivchenko, E. L.

M. M. Glazov and E. L. Ivchenko, “Spin noise in quantum dot ensembles,” Phys. Rev. B 86, 115308 (2012).
[Crossref]

E. L. Ivchenko, Optical Spectroscopy of Semiconductor Nanostructures (Alpha Science, 2005).

Jeltes, T.

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury BrownTwiss effect for bosons and fermions,” Nature 445, 402 (2007).
[Crossref] [PubMed]

Johnson, P. O.

A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691 (1955).
[Crossref]

Koch, S. W.

T. Meier, P. Thomas, and S. W. Koch, Coherent Semiconductor Optics: From Basic Concepts to Nanostructure Applications (Springer, 2007).
[Crossref]

Koller, C.

A. Perrin, R. Bücker, S. Manz, T. Betz, C. Koller, T. Plisson, T. Schumm, and J. Schmiedmayer, “Hanbury Brown and Twiss correlations across the BoseEinstein condensation threshold,” Nat. Phys. 8, 195 (2012).
[Crossref]

Kozlov, G. G.

G. G. Kozlov and V. S. Zapasskii, “Intensity-related optical susceptibility and ‘active’ noise spectroscopy,” J. of Phys. B 47, 185401 (2014).
[Crossref]

V. S. Zapasskii, A. Greilich, S. A. Crooker, Y. Li, G. G. Kozlov, D. R. Yakovlev, D. Reuter, A. D. Wieck, and M. Bayer, “Optical spectroscopy of spin noise,” Phys. Rev. Lett. 110, 176601 (2013).
[Crossref] [PubMed]

V. S. Zapasskii and G. G. Kozlov, “Slow light and slow current,” Opt. Spectrosc. 104, 95 (2008).
[Crossref]

Krachmalnicoff, V.

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury BrownTwiss effect for bosons and fermions,” Nature 445, 402 (2007).
[Crossref] [PubMed]

Lahini, Y

Y Lahini, Y Bromberg, Y. Shechtman, A. Szameit, D. N. Christodoulides, R. Morandotti, and Y Silberberg, “Hanbury-Brown and Twiss correlations of Anderson localized waves,” Phys. Rev. A 84, 041806 (2011).
[Crossref]

Lahini, Y.

Y Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury-Brown and Twiss interferometry with interacting photons,” Nat. Photonics 4, 721 (2010).
[Crossref]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Butterworth-Heinemann, 2004).

Li, Y.

V. S. Zapasskii, A. Greilich, S. A. Crooker, Y. Li, G. G. Kozlov, D. R. Yakovlev, D. Reuter, A. D. Wieck, and M. Bayer, “Optical spectroscopy of spin noise,” Phys. Rev. Lett. 110, 176601 (2013).
[Crossref] [PubMed]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Butterworth-Heinemann, 2004).

Lomakin, A. V.

E. B. Aleksandrov, Yu. M. Golubev, A. V. Lomakin, and V. A. Noskin, “Intensity-fluctuation spectroscopy of optical fields with non-Gaussian statistics,” Sov. Phys. Usp. 26, 643 (1983).
[Crossref]

Lyssenkot, V. G.

J. Erland, K.-H. Pantke, V. Mizeikis, V. G. Lyssenkot, and J. M. Hvam, “Spectrally resolved four-wave mixing in semiconductors: influence of inhomogeneous broadening,” Phys. Rev. B 5015047 (1994).
[Crossref]

Manz, S.

A. Perrin, R. Bücker, S. Manz, T. Betz, C. Koller, T. Plisson, T. Schumm, and J. Schmiedmayer, “Hanbury Brown and Twiss correlations across the BoseEinstein condensation threshold,” Nat. Phys. 8, 195 (2012).
[Crossref]

McNamara, J. M.

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury BrownTwiss effect for bosons and fermions,” Nature 445, 402 (2007).
[Crossref] [PubMed]

Meier, T.

T. Meier, P. Thomas, and S. W. Koch, Coherent Semiconductor Optics: From Basic Concepts to Nanostructure Applications (Springer, 2007).
[Crossref]

Mizeikis, V.

J. Erland, K.-H. Pantke, V. Mizeikis, V. G. Lyssenkot, and J. M. Hvam, “Spectrally resolved four-wave mixing in semiconductors: influence of inhomogeneous broadening,” Phys. Rev. B 5015047 (1994).
[Crossref]

Morandotti, R.

Y Lahini, Y Bromberg, Y. Shechtman, A. Szameit, D. N. Christodoulides, R. Morandotti, and Y Silberberg, “Hanbury-Brown and Twiss correlations of Anderson localized waves,” Phys. Rev. A 84, 041806 (2011).
[Crossref]

Müller, G. M.

G. M. Müller, M. Oestreich, M. Römer, and J. Hübner, “Semiconductor spin noise spectroscopy: fundamentals, accomplishments, and challenges,” Physica E 43, 569 (2010).
[Crossref]

Noskin, V. A.

E. B. Aleksandrov, Yu. M. Golubev, A. V. Lomakin, and V. A. Noskin, “Intensity-fluctuation spectroscopy of optical fields with non-Gaussian statistics,” Sov. Phys. Usp. 26, 643 (1983).
[Crossref]

Oestreich, M.

J. Hübner, F. Berski, R. Dahbashi, and M. Oestreich, “The rise of spin noise spectroscopy in semiconductors: from acoustic to GHz frequencies,” Phys. Stat. Solidi B 251, 1824 (2014).
[Crossref]

G. M. Müller, M. Oestreich, M. Römer, and J. Hübner, “Semiconductor spin noise spectroscopy: fundamentals, accomplishments, and challenges,” Physica E 43, 569 (2010).
[Crossref]

M. Römer, J. Hübner, and M. Oestreich, “Spatially resolved doping concentration measurement in semiconductors via spin noise spectroscopy,” Appl. Phys. Lett. 94, 112105 (2009).
[Crossref]

Pantke, K.-H.

J. Erland, K.-H. Pantke, V. Mizeikis, V. G. Lyssenkot, and J. M. Hvam, “Spectrally resolved four-wave mixing in semiconductors: influence of inhomogeneous broadening,” Phys. Rev. B 5015047 (1994).
[Crossref]

Perel, V. I.

B. M. Gorbovitskii and V. I. Perel, “Aleksandrov and Zapasskii experiment and the Raman effect,” Opt. Spectrosc. 54, 229 (1983).

Perrin, A.

A. Perrin, R. Bücker, S. Manz, T. Betz, C. Koller, T. Plisson, T. Schumm, and J. Schmiedmayer, “Hanbury Brown and Twiss correlations across the BoseEinstein condensation threshold,” Nat. Phys. 8, 195 (2012).
[Crossref]

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury BrownTwiss effect for bosons and fermions,” Nature 445, 402 (2007).
[Crossref] [PubMed]

Pershin, Yu. V.

Yu. V. Pershin, V. A. Slipko, D. Roy, and N. A. Sinitsyn, “Two-beam spin noise spectroscopy,” Appl. Phys. Lett. 102, 202405 (2013).
[Crossref]

Plisson, T.

A. Perrin, R. Bücker, S. Manz, T. Betz, C. Koller, T. Plisson, T. Schumm, and J. Schmiedmayer, “Hanbury Brown and Twiss correlations across the BoseEinstein condensation threshold,” Nat. Phys. 8, 195 (2012).
[Crossref]

Pound, R.V.

G.A. Rebka and R.V. Pound, “Time-correlated photons,” Nature 180, 1035 (1957).
[Crossref]

Rebka, G.A.

G.A. Rebka and R.V. Pound, “Time-correlated photons,” Nature 180, 1035 (1957).
[Crossref]

Reuter, D.

L. Yang, P. Glasenapp, A. Greilich, D. Reuter, A. D. Wieck, D. R. Yakovlev, M. Bayer, and S. A. Crooker, “Two-colour spin noise spectroscopy and fluctuation correlations reveal homogeneous linewidths within quantum-dot ensembles,” Nat. Commun. 5, 4949 (2014).
[Crossref] [PubMed]

V. S. Zapasskii, A. Greilich, S. A. Crooker, Y. Li, G. G. Kozlov, D. R. Yakovlev, D. Reuter, A. D. Wieck, and M. Bayer, “Optical spectroscopy of spin noise,” Phys. Rev. Lett. 110, 176601 (2013).
[Crossref] [PubMed]

Rickel, D. G.

S. A. Crooker, D. G. Rickel, A. V. Balatsky, and D. L. Smith, “Spectroscopy of spontaneous spin noise as a probe of spin dynamics and magnetic resonance,” Nature 431, 49 (2004).
[Crossref] [PubMed]

Römer, M.

G. M. Müller, M. Oestreich, M. Römer, and J. Hübner, “Semiconductor spin noise spectroscopy: fundamentals, accomplishments, and challenges,” Physica E 43, 569 (2010).
[Crossref]

M. Römer, J. Hübner, and M. Oestreich, “Spatially resolved doping concentration measurement in semiconductors via spin noise spectroscopy,” Appl. Phys. Lett. 94, 112105 (2009).
[Crossref]

Rottwitt, K.

K. Rottwitt and P. Tidemand-Lichtenberg, Nonlinear Optics: Principles and Applications (Taylor & Francis, 2014).

Roy, D.

Yu. V. Pershin, V. A. Slipko, D. Roy, and N. A. Sinitsyn, “Two-beam spin noise spectroscopy,” Appl. Phys. Lett. 102, 202405 (2013).
[Crossref]

D. Roy, L. Yang, S. A. Crooker, and N. A. Sinitsyn, “Cross-correlation spin noise spectroscopy of heterogeneous interacting spin systems,” arXiv:1408.5399v1 (2014).

Said, A. A.

Schellekens, M.

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury BrownTwiss effect for bosons and fermions,” Nature 445, 402 (2007).
[Crossref] [PubMed]

Schmiedmayer, J.

A. Perrin, R. Bücker, S. Manz, T. Betz, C. Koller, T. Plisson, T. Schumm, and J. Schmiedmayer, “Hanbury Brown and Twiss correlations across the BoseEinstein condensation threshold,” Nat. Phys. 8, 195 (2012).
[Crossref]

Schumm, T.

A. Perrin, R. Bücker, S. Manz, T. Betz, C. Koller, T. Plisson, T. Schumm, and J. Schmiedmayer, “Hanbury Brown and Twiss correlations across the BoseEinstein condensation threshold,” Nat. Phys. 8, 195 (2012).
[Crossref]

Shechtman, Y.

Y Lahini, Y Bromberg, Y. Shechtman, A. Szameit, D. N. Christodoulides, R. Morandotti, and Y Silberberg, “Hanbury-Brown and Twiss correlations of Anderson localized waves,” Phys. Rev. A 84, 041806 (2011).
[Crossref]

Sheik-Bahae, M.

Shen, Y.-R.

Y.-R. Shen, The Principles of Nonlinear Optics (Wiley-Interscience, 1984).

Silberberg, Y

Y Lahini, Y Bromberg, Y. Shechtman, A. Szameit, D. N. Christodoulides, R. Morandotti, and Y Silberberg, “Hanbury-Brown and Twiss correlations of Anderson localized waves,” Phys. Rev. A 84, 041806 (2011).
[Crossref]

Silberberg, Y.

Y Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury-Brown and Twiss interferometry with interacting photons,” Nat. Photonics 4, 721 (2010).
[Crossref]

Sillitto, R.M.

R.M. Sillitto, “Correlation between events in photon detectors,” Nature 179, 1127 (1957).
[Crossref]

Sinitsyn, N. A.

Yu. V. Pershin, V. A. Slipko, D. Roy, and N. A. Sinitsyn, “Two-beam spin noise spectroscopy,” Appl. Phys. Lett. 102, 202405 (2013).
[Crossref]

D. Roy, L. Yang, S. A. Crooker, and N. A. Sinitsyn, “Cross-correlation spin noise spectroscopy of heterogeneous interacting spin systems,” arXiv:1408.5399v1 (2014).

Slipko, V. A.

Yu. V. Pershin, V. A. Slipko, D. Roy, and N. A. Sinitsyn, “Two-beam spin noise spectroscopy,” Appl. Phys. Lett. 102, 202405 (2013).
[Crossref]

Small, E.

Y Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury-Brown and Twiss interferometry with interacting photons,” Nat. Photonics 4, 721 (2010).
[Crossref]

Smith, D. L.

S. A. Crooker, D. G. Rickel, A. V. Balatsky, and D. L. Smith, “Spectroscopy of spontaneous spin noise as a probe of spin dynamics and magnetic resonance,” Nature 431, 49 (2004).
[Crossref] [PubMed]

Szameit, A.

Y Lahini, Y Bromberg, Y. Shechtman, A. Szameit, D. N. Christodoulides, R. Morandotti, and Y Silberberg, “Hanbury-Brown and Twiss correlations of Anderson localized waves,” Phys. Rev. A 84, 041806 (2011).
[Crossref]

Thomas, P.

T. Meier, P. Thomas, and S. W. Koch, Coherent Semiconductor Optics: From Basic Concepts to Nanostructure Applications (Springer, 2007).
[Crossref]

Tidemand-Lichtenberg, P.

K. Rottwitt and P. Tidemand-Lichtenberg, Nonlinear Optics: Principles and Applications (Taylor & Francis, 2014).

Twiss, R. Q.

R. Hanbury-Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27 (1956).
[Crossref]

Van Stryland, E. W.

E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[Crossref]

M. Sheik-Bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity, single-beam n2 measurements,” Opt. Lett. 14, 955 (1989).
[Crossref] [PubMed]

Vassen, W.

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury BrownTwiss effect for bosons and fermions,” Nature 445, 402 (2007).
[Crossref] [PubMed]

Westbrook, C. I.

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury BrownTwiss effect for bosons and fermions,” Nature 445, 402 (2007).
[Crossref] [PubMed]

Wieck, A. D.

L. Yang, P. Glasenapp, A. Greilich, D. Reuter, A. D. Wieck, D. R. Yakovlev, M. Bayer, and S. A. Crooker, “Two-colour spin noise spectroscopy and fluctuation correlations reveal homogeneous linewidths within quantum-dot ensembles,” Nat. Commun. 5, 4949 (2014).
[Crossref] [PubMed]

V. S. Zapasskii, A. Greilich, S. A. Crooker, Y. Li, G. G. Kozlov, D. R. Yakovlev, D. Reuter, A. D. Wieck, and M. Bayer, “Optical spectroscopy of spin noise,” Phys. Rev. Lett. 110, 176601 (2013).
[Crossref] [PubMed]

Yakovlev, D. R.

L. Yang, P. Glasenapp, A. Greilich, D. Reuter, A. D. Wieck, D. R. Yakovlev, M. Bayer, and S. A. Crooker, “Two-colour spin noise spectroscopy and fluctuation correlations reveal homogeneous linewidths within quantum-dot ensembles,” Nat. Commun. 5, 4949 (2014).
[Crossref] [PubMed]

V. S. Zapasskii, A. Greilich, S. A. Crooker, Y. Li, G. G. Kozlov, D. R. Yakovlev, D. Reuter, A. D. Wieck, and M. Bayer, “Optical spectroscopy of spin noise,” Phys. Rev. Lett. 110, 176601 (2013).
[Crossref] [PubMed]

Yang, L.

L. Yang, P. Glasenapp, A. Greilich, D. Reuter, A. D. Wieck, D. R. Yakovlev, M. Bayer, and S. A. Crooker, “Two-colour spin noise spectroscopy and fluctuation correlations reveal homogeneous linewidths within quantum-dot ensembles,” Nat. Commun. 5, 4949 (2014).
[Crossref] [PubMed]

D. Roy, L. Yang, S. A. Crooker, and N. A. Sinitsyn, “Cross-correlation spin noise spectroscopy of heterogeneous interacting spin systems,” arXiv:1408.5399v1 (2014).

Zapasskii, V.

E. Aleksandrov and V. Zapasskii, “Magnetic resonance in the Faraday-rotation noise spectrum,” J. Experim. Theor. Phys. 54, 64 (1981).

Zapasskii, V. S.

G. G. Kozlov and V. S. Zapasskii, “Intensity-related optical susceptibility and ‘active’ noise spectroscopy,” J. of Phys. B 47, 185401 (2014).
[Crossref]

V. S. Zapasskii, A. Greilich, S. A. Crooker, Y. Li, G. G. Kozlov, D. R. Yakovlev, D. Reuter, A. D. Wieck, and M. Bayer, “Optical spectroscopy of spin noise,” Phys. Rev. Lett. 110, 176601 (2013).
[Crossref] [PubMed]

V. S. Zapasskii, “Spin-noise spectroscopy: from proof of principle to applications,” Adv. Opt. Photon. 5, 131 (2013).
[Crossref]

V. S. Zapasskii and G. G. Kozlov, “Slow light and slow current,” Opt. Spectrosc. 104, 95 (2008).
[Crossref]

Adv. Opt. Photon. (1)

Annu. Rev. Phys. Chem. (1)

M. Dantus, “Coherent Nonlinear Spectroscopy: From Femtosecond Dynamics to Control,” Annu. Rev. Phys. Chem. 52, 639 (2001).
[Crossref] [PubMed]

Appl. Phys. Lett. (2)

M. Römer, J. Hübner, and M. Oestreich, “Spatially resolved doping concentration measurement in semiconductors via spin noise spectroscopy,” Appl. Phys. Lett. 94, 112105 (2009).
[Crossref]

Yu. V. Pershin, V. A. Slipko, D. Roy, and N. A. Sinitsyn, “Two-beam spin noise spectroscopy,” Appl. Phys. Lett. 102, 202405 (2013).
[Crossref]

IEEE J. Quantum Electron. (1)

E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[Crossref]

J. Experim. Theor. Phys. (1)

E. Aleksandrov and V. Zapasskii, “Magnetic resonance in the Faraday-rotation noise spectrum,” J. Experim. Theor. Phys. 54, 64 (1981).

J. of Phys. B (1)

G. G. Kozlov and V. S. Zapasskii, “Intensity-related optical susceptibility and ‘active’ noise spectroscopy,” J. of Phys. B 47, 185401 (2014).
[Crossref]

Nat. Commun. (1)

L. Yang, P. Glasenapp, A. Greilich, D. Reuter, A. D. Wieck, D. R. Yakovlev, M. Bayer, and S. A. Crooker, “Two-colour spin noise spectroscopy and fluctuation correlations reveal homogeneous linewidths within quantum-dot ensembles,” Nat. Commun. 5, 4949 (2014).
[Crossref] [PubMed]

Nat. Photonics (1)

Y Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury-Brown and Twiss interferometry with interacting photons,” Nat. Photonics 4, 721 (2010).
[Crossref]

Nat. Phys. (1)

A. Perrin, R. Bücker, S. Manz, T. Betz, C. Koller, T. Plisson, T. Schumm, and J. Schmiedmayer, “Hanbury Brown and Twiss correlations across the BoseEinstein condensation threshold,” Nat. Phys. 8, 195 (2012).
[Crossref]

Nature (6)

T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury BrownTwiss effect for bosons and fermions,” Nature 445, 402 (2007).
[Crossref] [PubMed]

R. Hanbury-Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27 (1956).
[Crossref]

R.M. Sillitto, “Correlation between events in photon detectors,” Nature 179, 1127 (1957).
[Crossref]

P. Fellgett, “Question of correlation between photons in coherent beams of light,” Nature 179, 956 (1957).
[Crossref]

G.A. Rebka and R.V. Pound, “Time-correlated photons,” Nature 180, 1035 (1957).
[Crossref]

S. A. Crooker, D. G. Rickel, A. V. Balatsky, and D. L. Smith, “Spectroscopy of spontaneous spin noise as a probe of spin dynamics and magnetic resonance,” Nature 431, 49 (2004).
[Crossref] [PubMed]

Opt. Lett. (1)

Opt. Spectrosc. (2)

B. M. Gorbovitskii and V. I. Perel, “Aleksandrov and Zapasskii experiment and the Raman effect,” Opt. Spectrosc. 54, 229 (1983).

V. S. Zapasskii and G. G. Kozlov, “Slow light and slow current,” Opt. Spectrosc. 104, 95 (2008).
[Crossref]

Phys. Rev. (1)

A. T. Forrester, R. A. Gudmundsen, and P. O. Johnson, “Photoelectric mixing of incoherent light,” Phys. Rev. 99, 1691 (1955).
[Crossref]

Phys. Rev. A (1)

Y Lahini, Y Bromberg, Y. Shechtman, A. Szameit, D. N. Christodoulides, R. Morandotti, and Y Silberberg, “Hanbury-Brown and Twiss correlations of Anderson localized waves,” Phys. Rev. A 84, 041806 (2011).
[Crossref]

Phys. Rev. B (2)

M. M. Glazov and E. L. Ivchenko, “Spin noise in quantum dot ensembles,” Phys. Rev. B 86, 115308 (2012).
[Crossref]

J. Erland, K.-H. Pantke, V. Mizeikis, V. G. Lyssenkot, and J. M. Hvam, “Spectrally resolved four-wave mixing in semiconductors: influence of inhomogeneous broadening,” Phys. Rev. B 5015047 (1994).
[Crossref]

Phys. Rev. Lett. (1)

V. S. Zapasskii, A. Greilich, S. A. Crooker, Y. Li, G. G. Kozlov, D. R. Yakovlev, D. Reuter, A. D. Wieck, and M. Bayer, “Optical spectroscopy of spin noise,” Phys. Rev. Lett. 110, 176601 (2013).
[Crossref] [PubMed]

Phys. Solid State (1)

M. M. Glazov, “Coherent spin dynamics of electrons and excitons in nanostructures (a review),” Phys. Solid State 54, 1 (2012).
[Crossref]

Phys. Stat. Solidi B (1)

J. Hübner, F. Berski, R. Dahbashi, and M. Oestreich, “The rise of spin noise spectroscopy in semiconductors: from acoustic to GHz frequencies,” Phys. Stat. Solidi B 251, 1824 (2014).
[Crossref]

Physica E (1)

G. M. Müller, M. Oestreich, M. Römer, and J. Hübner, “Semiconductor spin noise spectroscopy: fundamentals, accomplishments, and challenges,” Physica E 43, 569 (2010).
[Crossref]

Sov. Phys. Usp. (1)

E. B. Aleksandrov, Yu. M. Golubev, A. V. Lomakin, and V. A. Noskin, “Intensity-fluctuation spectroscopy of optical fields with non-Gaussian statistics,” Sov. Phys. Usp. 26, 643 (1983).
[Crossref]

Other (15)

H. Carmichael, An Open System Approach to Quantum Optics (Springer-Verlag, 1993).

E. L. Ivchenko, Optical Spectroscopy of Semiconductor Nanostructures (Alpha Science, 2005).

T. Meier, P. Thomas, and S. W. Koch, Coherent Semiconductor Optics: From Basic Concepts to Nanostructure Applications (Springer, 2007).
[Crossref]

D. Roy, L. Yang, S. A. Crooker, and N. A. Sinitsyn, “Cross-correlation spin noise spectroscopy of heterogeneous interacting spin systems,” arXiv:1408.5399v1 (2014).

W. E. Moerner, ed., Persistent Spectral Hole-Burning: Science and Applications, (Springer-Verlag, 1988).
[Crossref]

The electric field at the detector is, as a rule, inhomogeneous due to, e.g., inhomogeneity of the sample, effects of diffraction, etc. The way how the signal is picked up and/or averaged in the particular setup may change the results quantitatively.

In the latter case the field propagating backwards contains the contribution ∝ E+ due to specular nonresonant reflection from the sample surface.

There is also an interference of P+0 and P+′, which is briefly discussed below.

There could be minor contributions if spins can freely propagate and go between the spots illuminated by the first and the second beams, cf. [24].

In inhomogeneous systems Eq. (18) should be modified following approaches of [16,18,19] to take the inhomogeneity into account.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Butterworth-Heinemann, 2004).

K. Rottwitt and P. Tidemand-Lichtenberg, Nonlinear Optics: Principles and Applications (Taylor & Francis, 2014).

N. Bloembergen, Nonlinear Optics (World Scientific, 1996).

R. W. Boyd, Nonlinear Optics (Academic press, 2003).

Y.-R. Shen, The Principles of Nonlinear Optics (Wiley-Interscience, 1984).

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Figures (3)

Fig. 1
Fig. 1

Spin noise spectroscopy is a nonperturbative optical method of detecting magnetic resonance with abilities unusual for linear optics. (a) Schematics of conventional SNS setup [12]. LP is the linear polarizer, S is the sample, HWP is the half-wave plate, PBS is the polarization beam splitter, BD is the balanced detector, FFT SA is the fast-Fouriertransform spectrum analyzer; (b) Z-scan test and 3D tomography arise from an effective dependence of spin-noise signal on the light power density [17]; (c) optical spectroscopy of spin noise allows penetration inside an inner structure of optical transitions [18]; (d) twocolor spin noise spectroscopy allows measuring homogeneous linewidth of optical transitions [19]; (e) two-beam spin noise spectroscopy allows measuring spin transfer [24].

Fig. 2
Fig. 2

Schematic representation of two experimental approaches to measuring spin-flip Raman scattering. Transmitted field is modulated by spin fluctuations and, correspondingly, acquires Raman-shifted spectral sidebands. SA is the spectrum analyzer and PD is the photodetector. In conventional Raman scattering experiments, SA is the optical spectrometer, in spin noise spectroscopy experiments SA analyzes the light intensity spectrum in the range of radio-frequencies.

Fig. 3
Fig. 3

Schematic energy diagram of the model four-level system under consideration. (a) Eigenstates of the system |Sx = ±1/2) in the field B parallel to x-axis. Double-headed arrows show the transition frequency, ω0, probe beam frequency, ω, and Larmor frequency Ω b = BB/ℏ, where g is the Landé factor; δ is the detuning. (b) Schematics of the same diagram in the basis of nonstationary states |Sz = ±1/2〉 monitored by the light beam. Blue curved arrows depict mixing of the states by the transverse magnetic field. Dashed ellipses show the components of the polarization P of the two-level system.

Equations (20)

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

P i = χ i j E j .
P ( E 1 + E 2 ) = P ( E 1 ) + P ( E 2 ) .
R ( I 1 + I 2 ) = R ( I 1 ) + R ( I 2 ) ,
P i = χ i j ( 1 ) E j + χ i j k ( 2 ) E j E k + χ i j k l ( 3 ) E j E k E l + ,
i P ˙ ± = ( ω 0 i γ ) P ± ( 1 / 2 S z ) | d | 2 E ± e i ω t ,
P ± = P ± 0 ( t ) + P ± ( t ) ,
P ± 0 ( t ) = i | d | 2 E ± 2 e i ω t i ( ω 0 ω ) + γ ,
P ± ( t ) = i | d | 2 E ± d Ω 2 π e i ω t i Ω t S ˜ z ( Ω ) i ( ω 0 ω Ω ) + γ ,
E + ( t ) = α E + e i ω t + β [ P + 0 ( t ) + P + ( t ) ] ,
I ( ω ) E + ( t ) E + * ( 0 ) e i ω t d t .
S ˜ z ( Ω 1 ) S ˜ z * ( Ω 2 ) = 2 π δ ( Ω 1 Ω 2 ) ( S z 2 ) Ω 1 ,
I s c ( ω ) | E + | 2 ( S z 2 ) Ω Γ 0 2 ( ω 0 ω ) 2 + γ 2 = | E + | 2 ( S z 2 ) Ω Γ 0 2 δ 2 + γ 2 .
( S z 2 ) Ω = N 4 ( τ s 1 + ( Ω Ω B ) 2 τ s 2 + τ s 1 + ( Ω + Ω B ) 2 τ s 2 ) ,
I a ( t ) 1 T a t T a / 2 t + T a / 2 | E ( t ) | 2 d t .
δ I ( t ) | E + | 2 Γ 0 d Ω 2 π [ e i Ω t S ˜ z ( Ω ) ω 0 ω Ω i γ ] .
( δ I ) Ω 2 | E + | 4 ( S z 2 ) Ω Γ 0 2 δ 2 ( δ 2 + γ 2 ) 2 .
δ I ( t ) | E + | 4 Γ 0 2 d Ω 2 π { e i Ω t S ˜ z ( Ω ) [ ( ω 0 ω Ω ) i γ ] [ ( ω 0 ω ) + i γ ] } ,
( δ I ) Ω 2 | E + | 4 ( S z 2 ) Ω Γ 0 4 ( δ 2 + γ 2 ) 2 .
d Ω 2 π [ e i Ω t S ˜ 1 , z ( Ω ) ω 0 ω 1 i γ ] and d Ω 2 π [ e i Ω t S ˜ 2 , z ( Ω ) ω 0 ω 2 i γ ] ,
( δ I 1 δ I 2 ) Ω ( S ˜ 1 , z S ˜ 2 , z ) Ω [ ( ω 0 ω 1 ) 2 + γ 2 ] [ ( ω 0 ω 2 ) 2 + γ 2 ] .

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