Abstract

A unified theory of time-domain and frequency-domain four-wave mixing processes, which is based on the nonlinear response function R(t3, t2, t1), is developed. The response function is expressed in terms of the four-point correlation function of the dipole operator F(τ1, τ2, τ3, τ4) and is evaluated explicitly for a stochastic model of line broadening that holds for any correlation time of the bath. Our results interpolate between the fast-modulation limit, in which the optical Bloch equations are valid, and the static limit of inhomogeneous line broadening. As an example of the relationship between time-domain and frequency-domain four-wave mixing, we compare the capabilities of steady-state and transient coherent anti-Stokes Raman spectroscopy experiments to probe the vibrational dynamics in ground and excited electronic states.

© 1986 Optical Society of America

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  1. N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).
  2. P. N. Butcher, Nonlinear Optical Phenomena (Ohio U. Press, Athens, Ohio, 1965).
  3. M. D. Levenson, Introduction to Nonlinear Laser Spectroscopy (Academic, New York, 1982).
  4. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  5. N. Bloembergen, H. Lotem, R. T. Lynch, “Lineshapes in coherent resonant Raman scattering,” Indian J. Pure Appl. Phys. 16, 151 (1978).
  6. S. A. J. Druet, J. P. E. Taran, “CARS spectroscopy,” Prog. Quantum Electron. 7, 1 (1981).
    [CrossRef]
  7. T. K. Lee, T. K. Gustafson, “Diagrammatic analysis of the density operator for nonlinear optical calculations: pulsed and cw responses,” Phys. Rev. A 18, 1597 (1978).
    [CrossRef]
  8. J. L. Oudar, Y. R. Shen, “Nonlinear spectroscopy by multi-resonant four-wave mixing,” Phys. Rev. A 22, 1141 (1980).
    [CrossRef]
  9. Y. Prior, A. R. Bogdan, M. Dagenais, N. Bloembergen, “Pressure-induced extra resonances in four-wave mixing,” Phys. Rev. Lett. 46, 111 (1981); A. R. Bogdan, M. W. Downer, N. Bloembergen, “Quantitative characteristics of pressure-induced four-wave mixing signals observed with cw laser beams,” Phys. Rev. A 24, 623 (1981); L. J. Rothberg, N. Bloembergen, “High resolution four-wave light mixing studies of collision-induced coherence in Na vapor,” Phys. Rev. A 30, 820 (1984).
    [CrossRef]
  10. J. R. Andrews, R. M. Hochstrasser, “Thermally induced excited-state coherent Raman spectra of solids,” Chem. Phys. Lett. 82, 381 (1981); J. R. Andrews, R. M. Hochstrasser, H. P. Trommsdorff, “Vibrational transitions in excited states of molecules using coherent Stokes Raman spectroscopy: application to ferrocytochrome-C,” Chem. Phys. 62, 87 (1981).
    [CrossRef]
  11. T. Yajima, H. Souma, “Study of ultra-fast relaxation processes by resonant Rayleigh-type optical mixing. I. Theory,” Phys. Rev. A 17, 309 (1978); T. Yajima, H. Souma, Y. Ishida, “Study of ultra-fast relaxation processes by resonant Rayleigh-type optical mixing. II. Experiment on dye solutions,” Phys. Rev. A 17, 324 (1978).
    [CrossRef]
  12. S. Mukamel, “Non-impact unified theory of four-wave mixing and two-photon processes,” Phys. Rev A 28, 3480 (1983).
    [CrossRef]
  13. V. Mizrahi, Y. Prior, S. Mukamel, “Single atom versus coherent pressure induced extra resonances in four-photon processes,” Opt. Lett. 8, 145 (1983); R. W. Boyd, S. Mukamel, “The origin of spectral holes in pump-probe studies of homogeneously broadened lines,” Phys. Rev. A 29, 1973 (1984).
    [CrossRef] [PubMed]
  14. I. D. Abella, N. A. Kurnit, S. R. Hartmann, “Photon echoes,” Phys. Rev. 141, 391 (1966); S. R. Hartmann, “Photon, spin, and Raman echoes,” IEEE J. Quantum Electron. 4, 802 (1968); T. W. Mossberg, R. Kachru, A. M. Flusberg, S. R. Hartmann, “Echoes in gaseous media: a generalized theory of rephasing phenomena,” Phys. Rev. A 20, 1976 (1979).
    [CrossRef]
  15. W. H. Hesselink, D. A. Wiersma, “Theory and experimental aspects of photon echoes in molecular solids,” in Modern Problems in Condensed Matter Sciences, V. M. Agranovich, A. A. Maradudin, eds. (North-Holland, Amsterdam, 1983), Vol. 4, p. 249.
  16. R. W. Olson, F. G. Patterson, H. W. H. Lee, M. D. Fayer, “Delocalized electronic excitations of pentacene dimers in a pterphenyl host: picosecond photon echo experiments,” Chem. Phys. Lett. 78, 403 (1981).
    [CrossRef]
  17. R. F. Loring, S. Mukamel, “Unified theory of photon echoes: the passage from inhomogeneous to homogeneous line broadening,” Chem. Phys. Lett. 114, 426 (1985).
    [CrossRef]
  18. J. R. Salcedo, A. E. Siegman, D. D. Dlott, M. D. Fayer, “Dynamics of energy transport in molecular crystals: the picosecond transient grating method,” Phys. Rev. Lett. 41, 131 (1978).
    [CrossRef]
  19. M. D. Fayer, “Dynamics of molecules in condensed phases: picosecond holographic grating experiments,” Ann. Rev. Phys. Chem. 33, 63 (1982).
    [CrossRef]
  20. P. F. Liao, L. M. Humphrey, D. M. Bloom, “Determination of upper limits for spatial energy diffusion in ruby,” Phys. Rev. B 10, 4145 (1979).
    [CrossRef]
  21. J. K. Tyminski, R. C. Powell, W. K. Zwicker, “Investigation of four-wave mixing in Ndx La1−x P5O14,” Phys. Rev. B 29, 6074 (1984).
    [CrossRef]
  22. H. J. Eichler, “Laser-induced grating phenomena,” Opt. Acta 24, 631 (1977).
    [CrossRef]
  23. A. von Jena, H. E. Lessing, “Theory of laser-induced amplitude and phase gratings including photoselection, orientational relaxation, and population kinetics,” Opt. Quantum Electron. 11, 419 (1979).
    [CrossRef]
  24. R. F. Loring, S. Mukamel, “Microscopic theory of the transient grating experiment,” J. Chem. Phys. 83, 4353 (1985); “Extra resonance in four-wave mixing as a probe of exciton dynamics: the steady-state analog of the transient grating,” J. Chem. Phys. 84, 1228 (1986).
    [CrossRef]
  25. N. Bloembergen, “The stimulated Raman effect,” Am. J. Phys, 35, 989 (1967).
    [CrossRef]
  26. A. Laubereau, W. Kaiser, “Vibrational dynamics of liquids and solids investigated by picosecond light pulses,” Rev. Mod. Phys. 50, 607 (1978); W. Zinth, H.-J. Polland, A. Laubereau, W. Kaiser, “New results on ultrafast coherent excitation of molecular vibrations in liquids,” Appl. Phys. B 26, 77 (1981).
    [CrossRef]
  27. S. M. George, A. L. Harris, M. Berg, C. B. Harris, “Picosecond studies of the temperature dependence of homogeneous and inhomogeneous linewidth broadening in liquid acetonitrile,” J. Chem. Phys. 80, 83 (1984).
    [CrossRef]
  28. R. F. Loring, S. Mukamel, “Selectivity in coherent transient Raman measurements of vibrational dephasing in liquids,” J. Chem. Phys. 83, 2116 (1985).
    [CrossRef]
  29. I. I. Abram, R. M. Hochstrasser, J. E. Kohl, M. G. Semack, D. White, “Coherence loss for vibrational and librational excitations in solid nitrogen,” J. Chem. Phys. 71, 153 (1979); F. Ho, W. S. Tsay, J. Trout, S. Velsko, R. M. Hochstrasser, “Picosecond time-resolved CARS in isotopically mixed crystals of benzene,” Chem. Phys. Lett. 97, 141 (1983); S. Velsko, J. Trout, R. M. Hochstrasser, “Quantum beating of vibrational factor group components in molecular solids,” J. Chem. Phys. 79, 2114 (1983).
    [CrossRef]
  30. E. L. Chronister, D. D. Dlott, “Vibrational energy transfer and localization in disordered solids by picosecond CARS spectroscopy,” J. Chem. Phys. 79, 5286 (1984); C. L. Schosser, D. D. Dlott, “A picosecond CARS study of vibron dynamics in molecular crystals: temperature dependence of homogeneous and inhomogeneous linewidths,” J. Chem. Phys. 80, 1394 (1984).
    [CrossRef]
  31. B. H. Hesp, D. A. Wiersma, “Vibrational relaxation in neat crystals of naphthalene by picosecond CARS,” Chem. Phys. Lett. 75, 423 (1980).
    [CrossRef]
  32. D. P. Weitekamp, K. Duppen, D. A. Wiersma, “Delayed four-wave mixing spectroscopy in molecular crystals: a non-perturbative approach,” Phys. Rev. A 27, 3089 (1983).
    [CrossRef]
  33. S. Mukamel, “Collisional broadening of spectral line shapes in two-photon and multiphoton processes,” Phys. Rep. 93, 1 (1982).
    [CrossRef]
  34. A. M. Weiner, S. DeSilvestri, E. P. Ippen, “Three-pulse scattering for femtosecond dephasing studies: theory and experiment,” J. Opt. Soc. Am. B 2, 654 (1985).
    [CrossRef]
  35. R. G. Breene, Theories of Spectral Line Shape (Wiley, New York, 1981).
  36. K. Burnett, “Collisional redistribution of radiation,” Phys. Rep. 118, 339 (1985).
    [CrossRef]
  37. N. Bloembergen, E. M. Purcell, R. V. Pound, “Relaxation effects in nuclear magnetic resonance absorption,” Phys. Rev. 73, 679 (1948).
    [CrossRef]
  38. P. W. Anderson, P. R. Weiss, “Exchange narrowing in paramagnetic resonance,” Rev. Mod. Phys. 25, 269 (1953).
    [CrossRef]
  39. R. Kubo, “A stochastic theory of line-shape and relaxation,” in Fluctuations, Relaxation and Resonance in Magnetic Systems,” D. ter Haar, ed. (Plenum, New York, 1962), p. 23.
  40. T. Takagahara, E. Hanamura, R. Kubo, “Stochastic models of intermediate state interaction in second order optical processes—stationary response. I,” J. Phys. Soc. Jpn. 43, 802 (1977); “Stochastic models of intermediate state interaction in second order optical processes—stationary response. II,” J. Phys. Soc. Jpn. 43, 811 (1977); “Stochastic models of intermediate state interaction in second order optical processes—transient response,” J. Phys. Soc. Jpn. 43, 1522 (1977).
    [CrossRef]
  41. S. Mukamel, “Stochastic theory of resonance Raman line shapes of polyatomic molecules in condensed phases,” J. Chem. Phys. 82, 5398 (1985).
    [CrossRef]
  42. E. Hanamura, “Stochastic theory of coherent optical transients,” J. Phys. Soc. Jpn. 52, 2258 (1983); “Stochastic theory of coherent optical transients. II. Free induction decay in Pr+3:LaF3,” J. Phys. Soc. Jpn. 52, 3678 (1983); H. Tsunetsugu, T. Taniguchi, E. Hanamura, “Exact solution for two level electronic system with frequency modulation under laser irradiation,” Solid State Commun. 52, 663 (1984).
    [CrossRef]
  43. M. Aihara, “Non-Markovian theory of nonlinear optical phenomena associated with the extremely fast relaxation in condensed matter,” Phys. Rev. B 25, 53 (1982).
    [CrossRef]
  44. G. J. Small, “Persistent nonphotochemical hole burning and the dephasing of impurity electronic transitions in organic glasses,” in Spectroscopy and Excitation Dynamics of Condensed Molecular Systems, V. M. Agranovich, R. M. Hochstrasser, eds. (North-Holland, New York, 1983), pp. 515; T. C. Caau, C. K. Johnson, G. J. Small, “Multiresonant four-wave mixing spectroscopy of pentacene in naphthalene,” J. Phys. Chem. 89, 2984 (1985).
    [CrossRef]
  45. B. S. Hudson, W. H. Hetherington, S. P. Cramer, I. Chabay, G. K. Klauminzer, “Resonance enhanced coherent anti-Stokes Raman scattering,” Proc. Natl. Acad. Sci. (USA) 73, 3798 (1976).
    [CrossRef]
  46. L. A. Carreira, T. C. Maguire, T. B. Malloy, “Excitation profiles of the coherent anti-Stokes resonance Raman spectrum of β-carotene,” J. Chem. Phys. 66, 2621 (1977).
    [CrossRef]

1985

R. F. Loring, S. Mukamel, “Unified theory of photon echoes: the passage from inhomogeneous to homogeneous line broadening,” Chem. Phys. Lett. 114, 426 (1985).
[CrossRef]

R. F. Loring, S. Mukamel, “Microscopic theory of the transient grating experiment,” J. Chem. Phys. 83, 4353 (1985); “Extra resonance in four-wave mixing as a probe of exciton dynamics: the steady-state analog of the transient grating,” J. Chem. Phys. 84, 1228 (1986).
[CrossRef]

R. F. Loring, S. Mukamel, “Selectivity in coherent transient Raman measurements of vibrational dephasing in liquids,” J. Chem. Phys. 83, 2116 (1985).
[CrossRef]

A. M. Weiner, S. DeSilvestri, E. P. Ippen, “Three-pulse scattering for femtosecond dephasing studies: theory and experiment,” J. Opt. Soc. Am. B 2, 654 (1985).
[CrossRef]

K. Burnett, “Collisional redistribution of radiation,” Phys. Rep. 118, 339 (1985).
[CrossRef]

S. Mukamel, “Stochastic theory of resonance Raman line shapes of polyatomic molecules in condensed phases,” J. Chem. Phys. 82, 5398 (1985).
[CrossRef]

1984

E. L. Chronister, D. D. Dlott, “Vibrational energy transfer and localization in disordered solids by picosecond CARS spectroscopy,” J. Chem. Phys. 79, 5286 (1984); C. L. Schosser, D. D. Dlott, “A picosecond CARS study of vibron dynamics in molecular crystals: temperature dependence of homogeneous and inhomogeneous linewidths,” J. Chem. Phys. 80, 1394 (1984).
[CrossRef]

S. M. George, A. L. Harris, M. Berg, C. B. Harris, “Picosecond studies of the temperature dependence of homogeneous and inhomogeneous linewidth broadening in liquid acetonitrile,” J. Chem. Phys. 80, 83 (1984).
[CrossRef]

J. K. Tyminski, R. C. Powell, W. K. Zwicker, “Investigation of four-wave mixing in Ndx La1−x P5O14,” Phys. Rev. B 29, 6074 (1984).
[CrossRef]

1983

S. Mukamel, “Non-impact unified theory of four-wave mixing and two-photon processes,” Phys. Rev A 28, 3480 (1983).
[CrossRef]

V. Mizrahi, Y. Prior, S. Mukamel, “Single atom versus coherent pressure induced extra resonances in four-photon processes,” Opt. Lett. 8, 145 (1983); R. W. Boyd, S. Mukamel, “The origin of spectral holes in pump-probe studies of homogeneously broadened lines,” Phys. Rev. A 29, 1973 (1984).
[CrossRef] [PubMed]

E. Hanamura, “Stochastic theory of coherent optical transients,” J. Phys. Soc. Jpn. 52, 2258 (1983); “Stochastic theory of coherent optical transients. II. Free induction decay in Pr+3:LaF3,” J. Phys. Soc. Jpn. 52, 3678 (1983); H. Tsunetsugu, T. Taniguchi, E. Hanamura, “Exact solution for two level electronic system with frequency modulation under laser irradiation,” Solid State Commun. 52, 663 (1984).
[CrossRef]

D. P. Weitekamp, K. Duppen, D. A. Wiersma, “Delayed four-wave mixing spectroscopy in molecular crystals: a non-perturbative approach,” Phys. Rev. A 27, 3089 (1983).
[CrossRef]

1982

S. Mukamel, “Collisional broadening of spectral line shapes in two-photon and multiphoton processes,” Phys. Rep. 93, 1 (1982).
[CrossRef]

M. Aihara, “Non-Markovian theory of nonlinear optical phenomena associated with the extremely fast relaxation in condensed matter,” Phys. Rev. B 25, 53 (1982).
[CrossRef]

M. D. Fayer, “Dynamics of molecules in condensed phases: picosecond holographic grating experiments,” Ann. Rev. Phys. Chem. 33, 63 (1982).
[CrossRef]

1981

R. W. Olson, F. G. Patterson, H. W. H. Lee, M. D. Fayer, “Delocalized electronic excitations of pentacene dimers in a pterphenyl host: picosecond photon echo experiments,” Chem. Phys. Lett. 78, 403 (1981).
[CrossRef]

Y. Prior, A. R. Bogdan, M. Dagenais, N. Bloembergen, “Pressure-induced extra resonances in four-wave mixing,” Phys. Rev. Lett. 46, 111 (1981); A. R. Bogdan, M. W. Downer, N. Bloembergen, “Quantitative characteristics of pressure-induced four-wave mixing signals observed with cw laser beams,” Phys. Rev. A 24, 623 (1981); L. J. Rothberg, N. Bloembergen, “High resolution four-wave light mixing studies of collision-induced coherence in Na vapor,” Phys. Rev. A 30, 820 (1984).
[CrossRef]

J. R. Andrews, R. M. Hochstrasser, “Thermally induced excited-state coherent Raman spectra of solids,” Chem. Phys. Lett. 82, 381 (1981); J. R. Andrews, R. M. Hochstrasser, H. P. Trommsdorff, “Vibrational transitions in excited states of molecules using coherent Stokes Raman spectroscopy: application to ferrocytochrome-C,” Chem. Phys. 62, 87 (1981).
[CrossRef]

S. A. J. Druet, J. P. E. Taran, “CARS spectroscopy,” Prog. Quantum Electron. 7, 1 (1981).
[CrossRef]

1980

J. L. Oudar, Y. R. Shen, “Nonlinear spectroscopy by multi-resonant four-wave mixing,” Phys. Rev. A 22, 1141 (1980).
[CrossRef]

B. H. Hesp, D. A. Wiersma, “Vibrational relaxation in neat crystals of naphthalene by picosecond CARS,” Chem. Phys. Lett. 75, 423 (1980).
[CrossRef]

1979

I. I. Abram, R. M. Hochstrasser, J. E. Kohl, M. G. Semack, D. White, “Coherence loss for vibrational and librational excitations in solid nitrogen,” J. Chem. Phys. 71, 153 (1979); F. Ho, W. S. Tsay, J. Trout, S. Velsko, R. M. Hochstrasser, “Picosecond time-resolved CARS in isotopically mixed crystals of benzene,” Chem. Phys. Lett. 97, 141 (1983); S. Velsko, J. Trout, R. M. Hochstrasser, “Quantum beating of vibrational factor group components in molecular solids,” J. Chem. Phys. 79, 2114 (1983).
[CrossRef]

A. von Jena, H. E. Lessing, “Theory of laser-induced amplitude and phase gratings including photoselection, orientational relaxation, and population kinetics,” Opt. Quantum Electron. 11, 419 (1979).
[CrossRef]

P. F. Liao, L. M. Humphrey, D. M. Bloom, “Determination of upper limits for spatial energy diffusion in ruby,” Phys. Rev. B 10, 4145 (1979).
[CrossRef]

1978

J. R. Salcedo, A. E. Siegman, D. D. Dlott, M. D. Fayer, “Dynamics of energy transport in molecular crystals: the picosecond transient grating method,” Phys. Rev. Lett. 41, 131 (1978).
[CrossRef]

N. Bloembergen, H. Lotem, R. T. Lynch, “Lineshapes in coherent resonant Raman scattering,” Indian J. Pure Appl. Phys. 16, 151 (1978).

T. K. Lee, T. K. Gustafson, “Diagrammatic analysis of the density operator for nonlinear optical calculations: pulsed and cw responses,” Phys. Rev. A 18, 1597 (1978).
[CrossRef]

T. Yajima, H. Souma, “Study of ultra-fast relaxation processes by resonant Rayleigh-type optical mixing. I. Theory,” Phys. Rev. A 17, 309 (1978); T. Yajima, H. Souma, Y. Ishida, “Study of ultra-fast relaxation processes by resonant Rayleigh-type optical mixing. II. Experiment on dye solutions,” Phys. Rev. A 17, 324 (1978).
[CrossRef]

A. Laubereau, W. Kaiser, “Vibrational dynamics of liquids and solids investigated by picosecond light pulses,” Rev. Mod. Phys. 50, 607 (1978); W. Zinth, H.-J. Polland, A. Laubereau, W. Kaiser, “New results on ultrafast coherent excitation of molecular vibrations in liquids,” Appl. Phys. B 26, 77 (1981).
[CrossRef]

1977

H. J. Eichler, “Laser-induced grating phenomena,” Opt. Acta 24, 631 (1977).
[CrossRef]

T. Takagahara, E. Hanamura, R. Kubo, “Stochastic models of intermediate state interaction in second order optical processes—stationary response. I,” J. Phys. Soc. Jpn. 43, 802 (1977); “Stochastic models of intermediate state interaction in second order optical processes—stationary response. II,” J. Phys. Soc. Jpn. 43, 811 (1977); “Stochastic models of intermediate state interaction in second order optical processes—transient response,” J. Phys. Soc. Jpn. 43, 1522 (1977).
[CrossRef]

L. A. Carreira, T. C. Maguire, T. B. Malloy, “Excitation profiles of the coherent anti-Stokes resonance Raman spectrum of β-carotene,” J. Chem. Phys. 66, 2621 (1977).
[CrossRef]

1976

B. S. Hudson, W. H. Hetherington, S. P. Cramer, I. Chabay, G. K. Klauminzer, “Resonance enhanced coherent anti-Stokes Raman scattering,” Proc. Natl. Acad. Sci. (USA) 73, 3798 (1976).
[CrossRef]

1967

N. Bloembergen, “The stimulated Raman effect,” Am. J. Phys, 35, 989 (1967).
[CrossRef]

1966

I. D. Abella, N. A. Kurnit, S. R. Hartmann, “Photon echoes,” Phys. Rev. 141, 391 (1966); S. R. Hartmann, “Photon, spin, and Raman echoes,” IEEE J. Quantum Electron. 4, 802 (1968); T. W. Mossberg, R. Kachru, A. M. Flusberg, S. R. Hartmann, “Echoes in gaseous media: a generalized theory of rephasing phenomena,” Phys. Rev. A 20, 1976 (1979).
[CrossRef]

1953

P. W. Anderson, P. R. Weiss, “Exchange narrowing in paramagnetic resonance,” Rev. Mod. Phys. 25, 269 (1953).
[CrossRef]

1948

N. Bloembergen, E. M. Purcell, R. V. Pound, “Relaxation effects in nuclear magnetic resonance absorption,” Phys. Rev. 73, 679 (1948).
[CrossRef]

Abella, I. D.

I. D. Abella, N. A. Kurnit, S. R. Hartmann, “Photon echoes,” Phys. Rev. 141, 391 (1966); S. R. Hartmann, “Photon, spin, and Raman echoes,” IEEE J. Quantum Electron. 4, 802 (1968); T. W. Mossberg, R. Kachru, A. M. Flusberg, S. R. Hartmann, “Echoes in gaseous media: a generalized theory of rephasing phenomena,” Phys. Rev. A 20, 1976 (1979).
[CrossRef]

Abram, I. I.

I. I. Abram, R. M. Hochstrasser, J. E. Kohl, M. G. Semack, D. White, “Coherence loss for vibrational and librational excitations in solid nitrogen,” J. Chem. Phys. 71, 153 (1979); F. Ho, W. S. Tsay, J. Trout, S. Velsko, R. M. Hochstrasser, “Picosecond time-resolved CARS in isotopically mixed crystals of benzene,” Chem. Phys. Lett. 97, 141 (1983); S. Velsko, J. Trout, R. M. Hochstrasser, “Quantum beating of vibrational factor group components in molecular solids,” J. Chem. Phys. 79, 2114 (1983).
[CrossRef]

Aihara, M.

M. Aihara, “Non-Markovian theory of nonlinear optical phenomena associated with the extremely fast relaxation in condensed matter,” Phys. Rev. B 25, 53 (1982).
[CrossRef]

Anderson, P. W.

P. W. Anderson, P. R. Weiss, “Exchange narrowing in paramagnetic resonance,” Rev. Mod. Phys. 25, 269 (1953).
[CrossRef]

Andrews, J. R.

J. R. Andrews, R. M. Hochstrasser, “Thermally induced excited-state coherent Raman spectra of solids,” Chem. Phys. Lett. 82, 381 (1981); J. R. Andrews, R. M. Hochstrasser, H. P. Trommsdorff, “Vibrational transitions in excited states of molecules using coherent Stokes Raman spectroscopy: application to ferrocytochrome-C,” Chem. Phys. 62, 87 (1981).
[CrossRef]

Berg, M.

S. M. George, A. L. Harris, M. Berg, C. B. Harris, “Picosecond studies of the temperature dependence of homogeneous and inhomogeneous linewidth broadening in liquid acetonitrile,” J. Chem. Phys. 80, 83 (1984).
[CrossRef]

Bloembergen, N.

Y. Prior, A. R. Bogdan, M. Dagenais, N. Bloembergen, “Pressure-induced extra resonances in four-wave mixing,” Phys. Rev. Lett. 46, 111 (1981); A. R. Bogdan, M. W. Downer, N. Bloembergen, “Quantitative characteristics of pressure-induced four-wave mixing signals observed with cw laser beams,” Phys. Rev. A 24, 623 (1981); L. J. Rothberg, N. Bloembergen, “High resolution four-wave light mixing studies of collision-induced coherence in Na vapor,” Phys. Rev. A 30, 820 (1984).
[CrossRef]

N. Bloembergen, H. Lotem, R. T. Lynch, “Lineshapes in coherent resonant Raman scattering,” Indian J. Pure Appl. Phys. 16, 151 (1978).

N. Bloembergen, “The stimulated Raman effect,” Am. J. Phys, 35, 989 (1967).
[CrossRef]

N. Bloembergen, E. M. Purcell, R. V. Pound, “Relaxation effects in nuclear magnetic resonance absorption,” Phys. Rev. 73, 679 (1948).
[CrossRef]

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

Bloom, D. M.

P. F. Liao, L. M. Humphrey, D. M. Bloom, “Determination of upper limits for spatial energy diffusion in ruby,” Phys. Rev. B 10, 4145 (1979).
[CrossRef]

Bogdan, A. R.

Y. Prior, A. R. Bogdan, M. Dagenais, N. Bloembergen, “Pressure-induced extra resonances in four-wave mixing,” Phys. Rev. Lett. 46, 111 (1981); A. R. Bogdan, M. W. Downer, N. Bloembergen, “Quantitative characteristics of pressure-induced four-wave mixing signals observed with cw laser beams,” Phys. Rev. A 24, 623 (1981); L. J. Rothberg, N. Bloembergen, “High resolution four-wave light mixing studies of collision-induced coherence in Na vapor,” Phys. Rev. A 30, 820 (1984).
[CrossRef]

Breene, R. G.

R. G. Breene, Theories of Spectral Line Shape (Wiley, New York, 1981).

Burnett, K.

K. Burnett, “Collisional redistribution of radiation,” Phys. Rep. 118, 339 (1985).
[CrossRef]

Butcher, P. N.

P. N. Butcher, Nonlinear Optical Phenomena (Ohio U. Press, Athens, Ohio, 1965).

Carreira, L. A.

L. A. Carreira, T. C. Maguire, T. B. Malloy, “Excitation profiles of the coherent anti-Stokes resonance Raman spectrum of β-carotene,” J. Chem. Phys. 66, 2621 (1977).
[CrossRef]

Chabay, I.

B. S. Hudson, W. H. Hetherington, S. P. Cramer, I. Chabay, G. K. Klauminzer, “Resonance enhanced coherent anti-Stokes Raman scattering,” Proc. Natl. Acad. Sci. (USA) 73, 3798 (1976).
[CrossRef]

Chronister, E. L.

E. L. Chronister, D. D. Dlott, “Vibrational energy transfer and localization in disordered solids by picosecond CARS spectroscopy,” J. Chem. Phys. 79, 5286 (1984); C. L. Schosser, D. D. Dlott, “A picosecond CARS study of vibron dynamics in molecular crystals: temperature dependence of homogeneous and inhomogeneous linewidths,” J. Chem. Phys. 80, 1394 (1984).
[CrossRef]

Cramer, S. P.

B. S. Hudson, W. H. Hetherington, S. P. Cramer, I. Chabay, G. K. Klauminzer, “Resonance enhanced coherent anti-Stokes Raman scattering,” Proc. Natl. Acad. Sci. (USA) 73, 3798 (1976).
[CrossRef]

Dagenais, M.

Y. Prior, A. R. Bogdan, M. Dagenais, N. Bloembergen, “Pressure-induced extra resonances in four-wave mixing,” Phys. Rev. Lett. 46, 111 (1981); A. R. Bogdan, M. W. Downer, N. Bloembergen, “Quantitative characteristics of pressure-induced four-wave mixing signals observed with cw laser beams,” Phys. Rev. A 24, 623 (1981); L. J. Rothberg, N. Bloembergen, “High resolution four-wave light mixing studies of collision-induced coherence in Na vapor,” Phys. Rev. A 30, 820 (1984).
[CrossRef]

DeSilvestri, S.

Dlott, D. D.

E. L. Chronister, D. D. Dlott, “Vibrational energy transfer and localization in disordered solids by picosecond CARS spectroscopy,” J. Chem. Phys. 79, 5286 (1984); C. L. Schosser, D. D. Dlott, “A picosecond CARS study of vibron dynamics in molecular crystals: temperature dependence of homogeneous and inhomogeneous linewidths,” J. Chem. Phys. 80, 1394 (1984).
[CrossRef]

J. R. Salcedo, A. E. Siegman, D. D. Dlott, M. D. Fayer, “Dynamics of energy transport in molecular crystals: the picosecond transient grating method,” Phys. Rev. Lett. 41, 131 (1978).
[CrossRef]

Druet, S. A. J.

S. A. J. Druet, J. P. E. Taran, “CARS spectroscopy,” Prog. Quantum Electron. 7, 1 (1981).
[CrossRef]

Duppen, K.

D. P. Weitekamp, K. Duppen, D. A. Wiersma, “Delayed four-wave mixing spectroscopy in molecular crystals: a non-perturbative approach,” Phys. Rev. A 27, 3089 (1983).
[CrossRef]

Eichler, H. J.

H. J. Eichler, “Laser-induced grating phenomena,” Opt. Acta 24, 631 (1977).
[CrossRef]

Fayer, M. D.

M. D. Fayer, “Dynamics of molecules in condensed phases: picosecond holographic grating experiments,” Ann. Rev. Phys. Chem. 33, 63 (1982).
[CrossRef]

R. W. Olson, F. G. Patterson, H. W. H. Lee, M. D. Fayer, “Delocalized electronic excitations of pentacene dimers in a pterphenyl host: picosecond photon echo experiments,” Chem. Phys. Lett. 78, 403 (1981).
[CrossRef]

J. R. Salcedo, A. E. Siegman, D. D. Dlott, M. D. Fayer, “Dynamics of energy transport in molecular crystals: the picosecond transient grating method,” Phys. Rev. Lett. 41, 131 (1978).
[CrossRef]

George, S. M.

S. M. George, A. L. Harris, M. Berg, C. B. Harris, “Picosecond studies of the temperature dependence of homogeneous and inhomogeneous linewidth broadening in liquid acetonitrile,” J. Chem. Phys. 80, 83 (1984).
[CrossRef]

Gustafson, T. K.

T. K. Lee, T. K. Gustafson, “Diagrammatic analysis of the density operator for nonlinear optical calculations: pulsed and cw responses,” Phys. Rev. A 18, 1597 (1978).
[CrossRef]

Hanamura, E.

E. Hanamura, “Stochastic theory of coherent optical transients,” J. Phys. Soc. Jpn. 52, 2258 (1983); “Stochastic theory of coherent optical transients. II. Free induction decay in Pr+3:LaF3,” J. Phys. Soc. Jpn. 52, 3678 (1983); H. Tsunetsugu, T. Taniguchi, E. Hanamura, “Exact solution for two level electronic system with frequency modulation under laser irradiation,” Solid State Commun. 52, 663 (1984).
[CrossRef]

T. Takagahara, E. Hanamura, R. Kubo, “Stochastic models of intermediate state interaction in second order optical processes—stationary response. I,” J. Phys. Soc. Jpn. 43, 802 (1977); “Stochastic models of intermediate state interaction in second order optical processes—stationary response. II,” J. Phys. Soc. Jpn. 43, 811 (1977); “Stochastic models of intermediate state interaction in second order optical processes—transient response,” J. Phys. Soc. Jpn. 43, 1522 (1977).
[CrossRef]

Harris, A. L.

S. M. George, A. L. Harris, M. Berg, C. B. Harris, “Picosecond studies of the temperature dependence of homogeneous and inhomogeneous linewidth broadening in liquid acetonitrile,” J. Chem. Phys. 80, 83 (1984).
[CrossRef]

Harris, C. B.

S. M. George, A. L. Harris, M. Berg, C. B. Harris, “Picosecond studies of the temperature dependence of homogeneous and inhomogeneous linewidth broadening in liquid acetonitrile,” J. Chem. Phys. 80, 83 (1984).
[CrossRef]

Hartmann, S. R.

I. D. Abella, N. A. Kurnit, S. R. Hartmann, “Photon echoes,” Phys. Rev. 141, 391 (1966); S. R. Hartmann, “Photon, spin, and Raman echoes,” IEEE J. Quantum Electron. 4, 802 (1968); T. W. Mossberg, R. Kachru, A. M. Flusberg, S. R. Hartmann, “Echoes in gaseous media: a generalized theory of rephasing phenomena,” Phys. Rev. A 20, 1976 (1979).
[CrossRef]

Hesp, B. H.

B. H. Hesp, D. A. Wiersma, “Vibrational relaxation in neat crystals of naphthalene by picosecond CARS,” Chem. Phys. Lett. 75, 423 (1980).
[CrossRef]

Hesselink, W. H.

W. H. Hesselink, D. A. Wiersma, “Theory and experimental aspects of photon echoes in molecular solids,” in Modern Problems in Condensed Matter Sciences, V. M. Agranovich, A. A. Maradudin, eds. (North-Holland, Amsterdam, 1983), Vol. 4, p. 249.

Hetherington, W. H.

B. S. Hudson, W. H. Hetherington, S. P. Cramer, I. Chabay, G. K. Klauminzer, “Resonance enhanced coherent anti-Stokes Raman scattering,” Proc. Natl. Acad. Sci. (USA) 73, 3798 (1976).
[CrossRef]

Hochstrasser, R. M.

J. R. Andrews, R. M. Hochstrasser, “Thermally induced excited-state coherent Raman spectra of solids,” Chem. Phys. Lett. 82, 381 (1981); J. R. Andrews, R. M. Hochstrasser, H. P. Trommsdorff, “Vibrational transitions in excited states of molecules using coherent Stokes Raman spectroscopy: application to ferrocytochrome-C,” Chem. Phys. 62, 87 (1981).
[CrossRef]

I. I. Abram, R. M. Hochstrasser, J. E. Kohl, M. G. Semack, D. White, “Coherence loss for vibrational and librational excitations in solid nitrogen,” J. Chem. Phys. 71, 153 (1979); F. Ho, W. S. Tsay, J. Trout, S. Velsko, R. M. Hochstrasser, “Picosecond time-resolved CARS in isotopically mixed crystals of benzene,” Chem. Phys. Lett. 97, 141 (1983); S. Velsko, J. Trout, R. M. Hochstrasser, “Quantum beating of vibrational factor group components in molecular solids,” J. Chem. Phys. 79, 2114 (1983).
[CrossRef]

Hudson, B. S.

B. S. Hudson, W. H. Hetherington, S. P. Cramer, I. Chabay, G. K. Klauminzer, “Resonance enhanced coherent anti-Stokes Raman scattering,” Proc. Natl. Acad. Sci. (USA) 73, 3798 (1976).
[CrossRef]

Humphrey, L. M.

P. F. Liao, L. M. Humphrey, D. M. Bloom, “Determination of upper limits for spatial energy diffusion in ruby,” Phys. Rev. B 10, 4145 (1979).
[CrossRef]

Ippen, E. P.

Kaiser, W.

A. Laubereau, W. Kaiser, “Vibrational dynamics of liquids and solids investigated by picosecond light pulses,” Rev. Mod. Phys. 50, 607 (1978); W. Zinth, H.-J. Polland, A. Laubereau, W. Kaiser, “New results on ultrafast coherent excitation of molecular vibrations in liquids,” Appl. Phys. B 26, 77 (1981).
[CrossRef]

Klauminzer, G. K.

B. S. Hudson, W. H. Hetherington, S. P. Cramer, I. Chabay, G. K. Klauminzer, “Resonance enhanced coherent anti-Stokes Raman scattering,” Proc. Natl. Acad. Sci. (USA) 73, 3798 (1976).
[CrossRef]

Kohl, J. E.

I. I. Abram, R. M. Hochstrasser, J. E. Kohl, M. G. Semack, D. White, “Coherence loss for vibrational and librational excitations in solid nitrogen,” J. Chem. Phys. 71, 153 (1979); F. Ho, W. S. Tsay, J. Trout, S. Velsko, R. M. Hochstrasser, “Picosecond time-resolved CARS in isotopically mixed crystals of benzene,” Chem. Phys. Lett. 97, 141 (1983); S. Velsko, J. Trout, R. M. Hochstrasser, “Quantum beating of vibrational factor group components in molecular solids,” J. Chem. Phys. 79, 2114 (1983).
[CrossRef]

Kubo, R.

T. Takagahara, E. Hanamura, R. Kubo, “Stochastic models of intermediate state interaction in second order optical processes—stationary response. I,” J. Phys. Soc. Jpn. 43, 802 (1977); “Stochastic models of intermediate state interaction in second order optical processes—stationary response. II,” J. Phys. Soc. Jpn. 43, 811 (1977); “Stochastic models of intermediate state interaction in second order optical processes—transient response,” J. Phys. Soc. Jpn. 43, 1522 (1977).
[CrossRef]

R. Kubo, “A stochastic theory of line-shape and relaxation,” in Fluctuations, Relaxation and Resonance in Magnetic Systems,” D. ter Haar, ed. (Plenum, New York, 1962), p. 23.

Kurnit, N. A.

I. D. Abella, N. A. Kurnit, S. R. Hartmann, “Photon echoes,” Phys. Rev. 141, 391 (1966); S. R. Hartmann, “Photon, spin, and Raman echoes,” IEEE J. Quantum Electron. 4, 802 (1968); T. W. Mossberg, R. Kachru, A. M. Flusberg, S. R. Hartmann, “Echoes in gaseous media: a generalized theory of rephasing phenomena,” Phys. Rev. A 20, 1976 (1979).
[CrossRef]

Laubereau, A.

A. Laubereau, W. Kaiser, “Vibrational dynamics of liquids and solids investigated by picosecond light pulses,” Rev. Mod. Phys. 50, 607 (1978); W. Zinth, H.-J. Polland, A. Laubereau, W. Kaiser, “New results on ultrafast coherent excitation of molecular vibrations in liquids,” Appl. Phys. B 26, 77 (1981).
[CrossRef]

Lee, H. W. H.

R. W. Olson, F. G. Patterson, H. W. H. Lee, M. D. Fayer, “Delocalized electronic excitations of pentacene dimers in a pterphenyl host: picosecond photon echo experiments,” Chem. Phys. Lett. 78, 403 (1981).
[CrossRef]

Lee, T. K.

T. K. Lee, T. K. Gustafson, “Diagrammatic analysis of the density operator for nonlinear optical calculations: pulsed and cw responses,” Phys. Rev. A 18, 1597 (1978).
[CrossRef]

Lessing, H. E.

A. von Jena, H. E. Lessing, “Theory of laser-induced amplitude and phase gratings including photoselection, orientational relaxation, and population kinetics,” Opt. Quantum Electron. 11, 419 (1979).
[CrossRef]

Levenson, M. D.

M. D. Levenson, Introduction to Nonlinear Laser Spectroscopy (Academic, New York, 1982).

Liao, P. F.

P. F. Liao, L. M. Humphrey, D. M. Bloom, “Determination of upper limits for spatial energy diffusion in ruby,” Phys. Rev. B 10, 4145 (1979).
[CrossRef]

Loring, R. F.

R. F. Loring, S. Mukamel, “Microscopic theory of the transient grating experiment,” J. Chem. Phys. 83, 4353 (1985); “Extra resonance in four-wave mixing as a probe of exciton dynamics: the steady-state analog of the transient grating,” J. Chem. Phys. 84, 1228 (1986).
[CrossRef]

R. F. Loring, S. Mukamel, “Selectivity in coherent transient Raman measurements of vibrational dephasing in liquids,” J. Chem. Phys. 83, 2116 (1985).
[CrossRef]

R. F. Loring, S. Mukamel, “Unified theory of photon echoes: the passage from inhomogeneous to homogeneous line broadening,” Chem. Phys. Lett. 114, 426 (1985).
[CrossRef]

Lotem, H.

N. Bloembergen, H. Lotem, R. T. Lynch, “Lineshapes in coherent resonant Raman scattering,” Indian J. Pure Appl. Phys. 16, 151 (1978).

Lynch, R. T.

N. Bloembergen, H. Lotem, R. T. Lynch, “Lineshapes in coherent resonant Raman scattering,” Indian J. Pure Appl. Phys. 16, 151 (1978).

Maguire, T. C.

L. A. Carreira, T. C. Maguire, T. B. Malloy, “Excitation profiles of the coherent anti-Stokes resonance Raman spectrum of β-carotene,” J. Chem. Phys. 66, 2621 (1977).
[CrossRef]

Malloy, T. B.

L. A. Carreira, T. C. Maguire, T. B. Malloy, “Excitation profiles of the coherent anti-Stokes resonance Raman spectrum of β-carotene,” J. Chem. Phys. 66, 2621 (1977).
[CrossRef]

Mizrahi, V.

Mukamel, S.

R. F. Loring, S. Mukamel, “Unified theory of photon echoes: the passage from inhomogeneous to homogeneous line broadening,” Chem. Phys. Lett. 114, 426 (1985).
[CrossRef]

R. F. Loring, S. Mukamel, “Selectivity in coherent transient Raman measurements of vibrational dephasing in liquids,” J. Chem. Phys. 83, 2116 (1985).
[CrossRef]

R. F. Loring, S. Mukamel, “Microscopic theory of the transient grating experiment,” J. Chem. Phys. 83, 4353 (1985); “Extra resonance in four-wave mixing as a probe of exciton dynamics: the steady-state analog of the transient grating,” J. Chem. Phys. 84, 1228 (1986).
[CrossRef]

S. Mukamel, “Stochastic theory of resonance Raman line shapes of polyatomic molecules in condensed phases,” J. Chem. Phys. 82, 5398 (1985).
[CrossRef]

S. Mukamel, “Non-impact unified theory of four-wave mixing and two-photon processes,” Phys. Rev A 28, 3480 (1983).
[CrossRef]

V. Mizrahi, Y. Prior, S. Mukamel, “Single atom versus coherent pressure induced extra resonances in four-photon processes,” Opt. Lett. 8, 145 (1983); R. W. Boyd, S. Mukamel, “The origin of spectral holes in pump-probe studies of homogeneously broadened lines,” Phys. Rev. A 29, 1973 (1984).
[CrossRef] [PubMed]

S. Mukamel, “Collisional broadening of spectral line shapes in two-photon and multiphoton processes,” Phys. Rep. 93, 1 (1982).
[CrossRef]

Olson, R. W.

R. W. Olson, F. G. Patterson, H. W. H. Lee, M. D. Fayer, “Delocalized electronic excitations of pentacene dimers in a pterphenyl host: picosecond photon echo experiments,” Chem. Phys. Lett. 78, 403 (1981).
[CrossRef]

Oudar, J. L.

J. L. Oudar, Y. R. Shen, “Nonlinear spectroscopy by multi-resonant four-wave mixing,” Phys. Rev. A 22, 1141 (1980).
[CrossRef]

Patterson, F. G.

R. W. Olson, F. G. Patterson, H. W. H. Lee, M. D. Fayer, “Delocalized electronic excitations of pentacene dimers in a pterphenyl host: picosecond photon echo experiments,” Chem. Phys. Lett. 78, 403 (1981).
[CrossRef]

Pound, R. V.

N. Bloembergen, E. M. Purcell, R. V. Pound, “Relaxation effects in nuclear magnetic resonance absorption,” Phys. Rev. 73, 679 (1948).
[CrossRef]

Powell, R. C.

J. K. Tyminski, R. C. Powell, W. K. Zwicker, “Investigation of four-wave mixing in Ndx La1−x P5O14,” Phys. Rev. B 29, 6074 (1984).
[CrossRef]

Prior, Y.

V. Mizrahi, Y. Prior, S. Mukamel, “Single atom versus coherent pressure induced extra resonances in four-photon processes,” Opt. Lett. 8, 145 (1983); R. W. Boyd, S. Mukamel, “The origin of spectral holes in pump-probe studies of homogeneously broadened lines,” Phys. Rev. A 29, 1973 (1984).
[CrossRef] [PubMed]

Y. Prior, A. R. Bogdan, M. Dagenais, N. Bloembergen, “Pressure-induced extra resonances in four-wave mixing,” Phys. Rev. Lett. 46, 111 (1981); A. R. Bogdan, M. W. Downer, N. Bloembergen, “Quantitative characteristics of pressure-induced four-wave mixing signals observed with cw laser beams,” Phys. Rev. A 24, 623 (1981); L. J. Rothberg, N. Bloembergen, “High resolution four-wave light mixing studies of collision-induced coherence in Na vapor,” Phys. Rev. A 30, 820 (1984).
[CrossRef]

Purcell, E. M.

N. Bloembergen, E. M. Purcell, R. V. Pound, “Relaxation effects in nuclear magnetic resonance absorption,” Phys. Rev. 73, 679 (1948).
[CrossRef]

Salcedo, J. R.

J. R. Salcedo, A. E. Siegman, D. D. Dlott, M. D. Fayer, “Dynamics of energy transport in molecular crystals: the picosecond transient grating method,” Phys. Rev. Lett. 41, 131 (1978).
[CrossRef]

Semack, M. G.

I. I. Abram, R. M. Hochstrasser, J. E. Kohl, M. G. Semack, D. White, “Coherence loss for vibrational and librational excitations in solid nitrogen,” J. Chem. Phys. 71, 153 (1979); F. Ho, W. S. Tsay, J. Trout, S. Velsko, R. M. Hochstrasser, “Picosecond time-resolved CARS in isotopically mixed crystals of benzene,” Chem. Phys. Lett. 97, 141 (1983); S. Velsko, J. Trout, R. M. Hochstrasser, “Quantum beating of vibrational factor group components in molecular solids,” J. Chem. Phys. 79, 2114 (1983).
[CrossRef]

Shen, Y. R.

J. L. Oudar, Y. R. Shen, “Nonlinear spectroscopy by multi-resonant four-wave mixing,” Phys. Rev. A 22, 1141 (1980).
[CrossRef]

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

Siegman, A. E.

J. R. Salcedo, A. E. Siegman, D. D. Dlott, M. D. Fayer, “Dynamics of energy transport in molecular crystals: the picosecond transient grating method,” Phys. Rev. Lett. 41, 131 (1978).
[CrossRef]

Small, G. J.

G. J. Small, “Persistent nonphotochemical hole burning and the dephasing of impurity electronic transitions in organic glasses,” in Spectroscopy and Excitation Dynamics of Condensed Molecular Systems, V. M. Agranovich, R. M. Hochstrasser, eds. (North-Holland, New York, 1983), pp. 515; T. C. Caau, C. K. Johnson, G. J. Small, “Multiresonant four-wave mixing spectroscopy of pentacene in naphthalene,” J. Phys. Chem. 89, 2984 (1985).
[CrossRef]

Souma, H.

T. Yajima, H. Souma, “Study of ultra-fast relaxation processes by resonant Rayleigh-type optical mixing. I. Theory,” Phys. Rev. A 17, 309 (1978); T. Yajima, H. Souma, Y. Ishida, “Study of ultra-fast relaxation processes by resonant Rayleigh-type optical mixing. II. Experiment on dye solutions,” Phys. Rev. A 17, 324 (1978).
[CrossRef]

Takagahara, T.

T. Takagahara, E. Hanamura, R. Kubo, “Stochastic models of intermediate state interaction in second order optical processes—stationary response. I,” J. Phys. Soc. Jpn. 43, 802 (1977); “Stochastic models of intermediate state interaction in second order optical processes—stationary response. II,” J. Phys. Soc. Jpn. 43, 811 (1977); “Stochastic models of intermediate state interaction in second order optical processes—transient response,” J. Phys. Soc. Jpn. 43, 1522 (1977).
[CrossRef]

Taran, J. P. E.

S. A. J. Druet, J. P. E. Taran, “CARS spectroscopy,” Prog. Quantum Electron. 7, 1 (1981).
[CrossRef]

Tyminski, J. K.

J. K. Tyminski, R. C. Powell, W. K. Zwicker, “Investigation of four-wave mixing in Ndx La1−x P5O14,” Phys. Rev. B 29, 6074 (1984).
[CrossRef]

von Jena, A.

A. von Jena, H. E. Lessing, “Theory of laser-induced amplitude and phase gratings including photoselection, orientational relaxation, and population kinetics,” Opt. Quantum Electron. 11, 419 (1979).
[CrossRef]

Weiner, A. M.

Weiss, P. R.

P. W. Anderson, P. R. Weiss, “Exchange narrowing in paramagnetic resonance,” Rev. Mod. Phys. 25, 269 (1953).
[CrossRef]

Weitekamp, D. P.

D. P. Weitekamp, K. Duppen, D. A. Wiersma, “Delayed four-wave mixing spectroscopy in molecular crystals: a non-perturbative approach,” Phys. Rev. A 27, 3089 (1983).
[CrossRef]

White, D.

I. I. Abram, R. M. Hochstrasser, J. E. Kohl, M. G. Semack, D. White, “Coherence loss for vibrational and librational excitations in solid nitrogen,” J. Chem. Phys. 71, 153 (1979); F. Ho, W. S. Tsay, J. Trout, S. Velsko, R. M. Hochstrasser, “Picosecond time-resolved CARS in isotopically mixed crystals of benzene,” Chem. Phys. Lett. 97, 141 (1983); S. Velsko, J. Trout, R. M. Hochstrasser, “Quantum beating of vibrational factor group components in molecular solids,” J. Chem. Phys. 79, 2114 (1983).
[CrossRef]

Wiersma, D. A.

D. P. Weitekamp, K. Duppen, D. A. Wiersma, “Delayed four-wave mixing spectroscopy in molecular crystals: a non-perturbative approach,” Phys. Rev. A 27, 3089 (1983).
[CrossRef]

B. H. Hesp, D. A. Wiersma, “Vibrational relaxation in neat crystals of naphthalene by picosecond CARS,” Chem. Phys. Lett. 75, 423 (1980).
[CrossRef]

W. H. Hesselink, D. A. Wiersma, “Theory and experimental aspects of photon echoes in molecular solids,” in Modern Problems in Condensed Matter Sciences, V. M. Agranovich, A. A. Maradudin, eds. (North-Holland, Amsterdam, 1983), Vol. 4, p. 249.

Yajima, T.

T. Yajima, H. Souma, “Study of ultra-fast relaxation processes by resonant Rayleigh-type optical mixing. I. Theory,” Phys. Rev. A 17, 309 (1978); T. Yajima, H. Souma, Y. Ishida, “Study of ultra-fast relaxation processes by resonant Rayleigh-type optical mixing. II. Experiment on dye solutions,” Phys. Rev. A 17, 324 (1978).
[CrossRef]

Zwicker, W. K.

J. K. Tyminski, R. C. Powell, W. K. Zwicker, “Investigation of four-wave mixing in Ndx La1−x P5O14,” Phys. Rev. B 29, 6074 (1984).
[CrossRef]

Am. J. Phys

N. Bloembergen, “The stimulated Raman effect,” Am. J. Phys, 35, 989 (1967).
[CrossRef]

Ann. Rev. Phys. Chem.

M. D. Fayer, “Dynamics of molecules in condensed phases: picosecond holographic grating experiments,” Ann. Rev. Phys. Chem. 33, 63 (1982).
[CrossRef]

Chem. Phys. Lett.

R. W. Olson, F. G. Patterson, H. W. H. Lee, M. D. Fayer, “Delocalized electronic excitations of pentacene dimers in a pterphenyl host: picosecond photon echo experiments,” Chem. Phys. Lett. 78, 403 (1981).
[CrossRef]

R. F. Loring, S. Mukamel, “Unified theory of photon echoes: the passage from inhomogeneous to homogeneous line broadening,” Chem. Phys. Lett. 114, 426 (1985).
[CrossRef]

J. R. Andrews, R. M. Hochstrasser, “Thermally induced excited-state coherent Raman spectra of solids,” Chem. Phys. Lett. 82, 381 (1981); J. R. Andrews, R. M. Hochstrasser, H. P. Trommsdorff, “Vibrational transitions in excited states of molecules using coherent Stokes Raman spectroscopy: application to ferrocytochrome-C,” Chem. Phys. 62, 87 (1981).
[CrossRef]

B. H. Hesp, D. A. Wiersma, “Vibrational relaxation in neat crystals of naphthalene by picosecond CARS,” Chem. Phys. Lett. 75, 423 (1980).
[CrossRef]

Indian J. Pure Appl. Phys.

N. Bloembergen, H. Lotem, R. T. Lynch, “Lineshapes in coherent resonant Raman scattering,” Indian J. Pure Appl. Phys. 16, 151 (1978).

J. Chem. Phys.

R. F. Loring, S. Mukamel, “Microscopic theory of the transient grating experiment,” J. Chem. Phys. 83, 4353 (1985); “Extra resonance in four-wave mixing as a probe of exciton dynamics: the steady-state analog of the transient grating,” J. Chem. Phys. 84, 1228 (1986).
[CrossRef]

S. M. George, A. L. Harris, M. Berg, C. B. Harris, “Picosecond studies of the temperature dependence of homogeneous and inhomogeneous linewidth broadening in liquid acetonitrile,” J. Chem. Phys. 80, 83 (1984).
[CrossRef]

R. F. Loring, S. Mukamel, “Selectivity in coherent transient Raman measurements of vibrational dephasing in liquids,” J. Chem. Phys. 83, 2116 (1985).
[CrossRef]

I. I. Abram, R. M. Hochstrasser, J. E. Kohl, M. G. Semack, D. White, “Coherence loss for vibrational and librational excitations in solid nitrogen,” J. Chem. Phys. 71, 153 (1979); F. Ho, W. S. Tsay, J. Trout, S. Velsko, R. M. Hochstrasser, “Picosecond time-resolved CARS in isotopically mixed crystals of benzene,” Chem. Phys. Lett. 97, 141 (1983); S. Velsko, J. Trout, R. M. Hochstrasser, “Quantum beating of vibrational factor group components in molecular solids,” J. Chem. Phys. 79, 2114 (1983).
[CrossRef]

E. L. Chronister, D. D. Dlott, “Vibrational energy transfer and localization in disordered solids by picosecond CARS spectroscopy,” J. Chem. Phys. 79, 5286 (1984); C. L. Schosser, D. D. Dlott, “A picosecond CARS study of vibron dynamics in molecular crystals: temperature dependence of homogeneous and inhomogeneous linewidths,” J. Chem. Phys. 80, 1394 (1984).
[CrossRef]

S. Mukamel, “Stochastic theory of resonance Raman line shapes of polyatomic molecules in condensed phases,” J. Chem. Phys. 82, 5398 (1985).
[CrossRef]

L. A. Carreira, T. C. Maguire, T. B. Malloy, “Excitation profiles of the coherent anti-Stokes resonance Raman spectrum of β-carotene,” J. Chem. Phys. 66, 2621 (1977).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. Soc. Jpn.

T. Takagahara, E. Hanamura, R. Kubo, “Stochastic models of intermediate state interaction in second order optical processes—stationary response. I,” J. Phys. Soc. Jpn. 43, 802 (1977); “Stochastic models of intermediate state interaction in second order optical processes—stationary response. II,” J. Phys. Soc. Jpn. 43, 811 (1977); “Stochastic models of intermediate state interaction in second order optical processes—transient response,” J. Phys. Soc. Jpn. 43, 1522 (1977).
[CrossRef]

E. Hanamura, “Stochastic theory of coherent optical transients,” J. Phys. Soc. Jpn. 52, 2258 (1983); “Stochastic theory of coherent optical transients. II. Free induction decay in Pr+3:LaF3,” J. Phys. Soc. Jpn. 52, 3678 (1983); H. Tsunetsugu, T. Taniguchi, E. Hanamura, “Exact solution for two level electronic system with frequency modulation under laser irradiation,” Solid State Commun. 52, 663 (1984).
[CrossRef]

Opt. Acta

H. J. Eichler, “Laser-induced grating phenomena,” Opt. Acta 24, 631 (1977).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

A. von Jena, H. E. Lessing, “Theory of laser-induced amplitude and phase gratings including photoselection, orientational relaxation, and population kinetics,” Opt. Quantum Electron. 11, 419 (1979).
[CrossRef]

Phys. Rep.

K. Burnett, “Collisional redistribution of radiation,” Phys. Rep. 118, 339 (1985).
[CrossRef]

S. Mukamel, “Collisional broadening of spectral line shapes in two-photon and multiphoton processes,” Phys. Rep. 93, 1 (1982).
[CrossRef]

Phys. Rev A

S. Mukamel, “Non-impact unified theory of four-wave mixing and two-photon processes,” Phys. Rev A 28, 3480 (1983).
[CrossRef]

Phys. Rev.

N. Bloembergen, E. M. Purcell, R. V. Pound, “Relaxation effects in nuclear magnetic resonance absorption,” Phys. Rev. 73, 679 (1948).
[CrossRef]

I. D. Abella, N. A. Kurnit, S. R. Hartmann, “Photon echoes,” Phys. Rev. 141, 391 (1966); S. R. Hartmann, “Photon, spin, and Raman echoes,” IEEE J. Quantum Electron. 4, 802 (1968); T. W. Mossberg, R. Kachru, A. M. Flusberg, S. R. Hartmann, “Echoes in gaseous media: a generalized theory of rephasing phenomena,” Phys. Rev. A 20, 1976 (1979).
[CrossRef]

Phys. Rev. A

T. Yajima, H. Souma, “Study of ultra-fast relaxation processes by resonant Rayleigh-type optical mixing. I. Theory,” Phys. Rev. A 17, 309 (1978); T. Yajima, H. Souma, Y. Ishida, “Study of ultra-fast relaxation processes by resonant Rayleigh-type optical mixing. II. Experiment on dye solutions,” Phys. Rev. A 17, 324 (1978).
[CrossRef]

T. K. Lee, T. K. Gustafson, “Diagrammatic analysis of the density operator for nonlinear optical calculations: pulsed and cw responses,” Phys. Rev. A 18, 1597 (1978).
[CrossRef]

J. L. Oudar, Y. R. Shen, “Nonlinear spectroscopy by multi-resonant four-wave mixing,” Phys. Rev. A 22, 1141 (1980).
[CrossRef]

D. P. Weitekamp, K. Duppen, D. A. Wiersma, “Delayed four-wave mixing spectroscopy in molecular crystals: a non-perturbative approach,” Phys. Rev. A 27, 3089 (1983).
[CrossRef]

Phys. Rev. B

P. F. Liao, L. M. Humphrey, D. M. Bloom, “Determination of upper limits for spatial energy diffusion in ruby,” Phys. Rev. B 10, 4145 (1979).
[CrossRef]

J. K. Tyminski, R. C. Powell, W. K. Zwicker, “Investigation of four-wave mixing in Ndx La1−x P5O14,” Phys. Rev. B 29, 6074 (1984).
[CrossRef]

M. Aihara, “Non-Markovian theory of nonlinear optical phenomena associated with the extremely fast relaxation in condensed matter,” Phys. Rev. B 25, 53 (1982).
[CrossRef]

Phys. Rev. Lett.

J. R. Salcedo, A. E. Siegman, D. D. Dlott, M. D. Fayer, “Dynamics of energy transport in molecular crystals: the picosecond transient grating method,” Phys. Rev. Lett. 41, 131 (1978).
[CrossRef]

Y. Prior, A. R. Bogdan, M. Dagenais, N. Bloembergen, “Pressure-induced extra resonances in four-wave mixing,” Phys. Rev. Lett. 46, 111 (1981); A. R. Bogdan, M. W. Downer, N. Bloembergen, “Quantitative characteristics of pressure-induced four-wave mixing signals observed with cw laser beams,” Phys. Rev. A 24, 623 (1981); L. J. Rothberg, N. Bloembergen, “High resolution four-wave light mixing studies of collision-induced coherence in Na vapor,” Phys. Rev. A 30, 820 (1984).
[CrossRef]

Proc. Natl. Acad. Sci. (USA)

B. S. Hudson, W. H. Hetherington, S. P. Cramer, I. Chabay, G. K. Klauminzer, “Resonance enhanced coherent anti-Stokes Raman scattering,” Proc. Natl. Acad. Sci. (USA) 73, 3798 (1976).
[CrossRef]

Prog. Quantum Electron.

S. A. J. Druet, J. P. E. Taran, “CARS spectroscopy,” Prog. Quantum Electron. 7, 1 (1981).
[CrossRef]

Rev. Mod. Phys.

A. Laubereau, W. Kaiser, “Vibrational dynamics of liquids and solids investigated by picosecond light pulses,” Rev. Mod. Phys. 50, 607 (1978); W. Zinth, H.-J. Polland, A. Laubereau, W. Kaiser, “New results on ultrafast coherent excitation of molecular vibrations in liquids,” Appl. Phys. B 26, 77 (1981).
[CrossRef]

P. W. Anderson, P. R. Weiss, “Exchange narrowing in paramagnetic resonance,” Rev. Mod. Phys. 25, 269 (1953).
[CrossRef]

Other

R. Kubo, “A stochastic theory of line-shape and relaxation,” in Fluctuations, Relaxation and Resonance in Magnetic Systems,” D. ter Haar, ed. (Plenum, New York, 1962), p. 23.

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

P. N. Butcher, Nonlinear Optical Phenomena (Ohio U. Press, Athens, Ohio, 1965).

M. D. Levenson, Introduction to Nonlinear Laser Spectroscopy (Academic, New York, 1982).

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

W. H. Hesselink, D. A. Wiersma, “Theory and experimental aspects of photon echoes in molecular solids,” in Modern Problems in Condensed Matter Sciences, V. M. Agranovich, A. A. Maradudin, eds. (North-Holland, Amsterdam, 1983), Vol. 4, p. 249.

G. J. Small, “Persistent nonphotochemical hole burning and the dephasing of impurity electronic transitions in organic glasses,” in Spectroscopy and Excitation Dynamics of Condensed Molecular Systems, V. M. Agranovich, R. M. Hochstrasser, eds. (North-Holland, New York, 1983), pp. 515; T. C. Caau, C. K. Johnson, G. J. Small, “Multiresonant four-wave mixing spectroscopy of pentacene in naphthalene,” J. Phys. Chem. 89, 2984 (1985).
[CrossRef]

R. G. Breene, Theories of Spectral Line Shape (Wiley, New York, 1981).

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Figures (5)

Fig. 1
Fig. 1

The general 4WM process. The coherent signal with wave vector ks is generated by a nonlinear mixing of the applied fields with wave vectors k1, k2, and k3.

Fig. 2
Fig. 2

The time arguments for Eqs. (12) and (16). The three radiative interactions occur at times τ1, τ2, and τ3, and the nonlinear polarization is calculated at time t. These time arguments are fully ordered: ττ2τ3t. t1, t2, and t3 denote the time intervals between the former time arguments as indicated in Eqs. (15).

Fig. 3
Fig. 3

The molecular level scheme and laser frequencies for 4WM. Levels |a〉 and |c〉 are part of the ground-state vibrational manifold, whereas levels |b〉 and |d〉 belong to an electronically excited manifold. γν is the inverse lifetime of level |ν〉. The electronic-dipole operator [Eq. (34)] couples vibronic states belonging to different electronic states.

Fig. 4
Fig. 4

Pictorial representation of the possible Liouville-space pathways12 that contribute to the nonlinear response function [Eq. (20)]. Solid lines denote radiative coupling V. Horizontal (vertical) lines represent action of V from the right (left). Starting at aa, after three perturbations the system finds itself along the dashed line. The dotted lines represent the last V, which acts from the left. At the end of four perturbations, the system is in a diagonal state (aa, bb, cc, or dd). There are, respectively, 1, 1, 3, and 3 three-bond pathways leading to aa, ba, dc, and cb. Altogether, there are therefore eight pathways. In each pathway each of the three fields acts once.

Fig. 5
Fig. 5

The eight distinct Liouville-space pathways that contribute to the nonlinear response function [Eq. (20) or (23)]. The eight terms in Eqs. (47), (48), (53a), and (58a) correspond, respectively, to, pathways (i)–(viii).

Equations (112)

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k s = ± k 1 ± k 2 ± k 3
ω s = ± ω 1 ± ω 2 ± ω 3 .
H T = H + H int .
H int ( t ) = α E ( r α , t ) V α ,
E ( r , t ) = j = 1 3 [ E j ( t ) exp ( i k j · r - i ω j t ) + E j * ( t ) exp ( - i k j · r + i ω j t ) ] .
H int ( t ) = E ( r , t ) V ,
ρ ( - ) = exp ( - β H ) / Tr exp ( - β H ) ,
d ρ d t = - i [ H , ρ ] - i [ H int , ρ ] .
d ρ d t = - i L ρ - i L int ρ .
L A [ H , A ] ,
L int A [ H int , A ] .
V A [ V , A ] .
P ( r , t ) V ρ ( t ) ,
A B Tr ( A B ) .
A L B Tr ( A L B ) .
P ( r , t ) = ( - i ) 3 - t d τ 3 - τ 3 d τ 2 - τ 2 d τ 1 V G ( t - τ 3 ) L int ( τ 3 ) × G ( τ 3 - τ 2 ) L int ( τ 2 ) G ( τ 2 - τ 1 ) L int ( τ 1 ) ρ ( - ) .
G ( τ ) exp ( - i L τ ) .
G ^ ( ω ) = - i 0 d τ exp ( i ω τ ) G ( τ ) = 1 ω - L .
t 1 = τ 2 - τ 1 , t 2 = τ 3 - τ 2 , t 3 = t - τ 3
τ 1 = t - t 1 - t 2 - t 3 , τ 2 = t - t 2 - t 3 , τ 3 = t - t 3 .
P ( r , t ) = ( - i ) 3 0 d t 1 0 d t 2 0 d t 3 × V G ( t 3 ) L int ( t - t 3 ) G ( t 2 ) L int ( t - t 2 - t 3 ) × G ( t 1 ) L int ( t - t 1 - t 2 - t 3 ) ρ ( - ) .
P ( r , t ) = k s , ω s exp ( i k s · r - i ω s t ) P ( k s , t ) .
k s = k 1 + k 2 + k 3
ω s = ω 1 + ω 2 + ω 3 .
P ( k s , t ) = ( - i ) 3 m , n , q = 1 , 2 , 3 0 d t 3 0 d t 2 0 d t 1 × R ( t 3 , t 2 , t 1 ) exp [ i ( ω m + ω n + ω q ) t 3 + i ( ω m + ω n ) t 2 + i ω m t 1 ] × E m ( t - t 1 - t 2 - t 3 ) E n ( t - t 2 - t 3 ) E q ( t - t 3 ) .
R ( t 3 , t 2 , t 1 ) V G ( t 3 ) V G ( t 2 ) V G ( t 1 ) V ρ ( - ) .
R ^ ( ω m + ω n + ω q , ω m + ω n , ω m ) ( - i ) 3 0 d t 3 0 d t 2 0 d t 1 × exp [ i ( ω m + ω n + ω q ) t 3 + i ( ω m + ω n ) t 2 + i ω m t 1 ] × R ( t 3 , t 2 , t 1 ) .
P ( k s , t ) = m , n , q = 1 , 2 , 3 - d ω m - d ω n - d ω q × R ^ ( ω m + ω n + ω q , ω m + ω n , ω m ) J m ( ω m ) J n ( ω n ) J q ( ω q ) × exp [ i ( ω m + ω n + ω q - ω m - ω n - ω q ) t ] ,
R ^ ( ω m + ω n + ω q , ω m + ω n , ω m ) = V G ^ ( ω m + ω n + ω q ) V G ^ ( ω m + ω n ) V G ^ ( ω m ) V ρ ( - ) ,
J j ( ω j ) = ( 2 π ) - 1 - d τ E j ( τ ) exp [ i ( ω j - ω j ) τ ] ,             j = m , n , q
S ( k s , t ) = P ( k s , t ) 2 .
E 1 ( τ ) = E 1 δ ( τ - τ * 1 ) , E 2 ( τ ) = E 2 δ ( τ - τ * 2 ) , E 3 ( τ ) = E 3 δ ( τ - τ * 3 ) ,
P s ( k s , t ) = E 1 E 2 E 3 R ( t 3 , t 2 , t 1 ) exp [ i ω 1 t 1 + i ( ω 1 + ω 2 ) t 2 + i ( ω 1 + ω 2 + ω 3 ) t 3 ]
S ( k s , t ) = E 1 E 2 E 3 2 R ( t 3 , t 2 , t 1 ) 2 .
J 1 ( ω 1 ) = E 1 δ ( ω 1 - ω 1 ) , J 2 ( ω 2 ) = E 2 δ ( ω 2 - ω 2 ) , J 3 ( ω 3 ) = E 3 δ ( ω 3 - ω 3 ) .
P ( k s , t ) = χ ( 3 ) ( - ω s , ω 1 , ω 2 , ω 3 ) E 1 E 2 E 3 ,
χ ( 3 ) ( - ω s , ω 1 , ω 2 , ω 3 ) = m , n , q = 1 , 2 , 3 R ^ ( ω m + ω n + ω q , ω m + ω n , ω m ) .
S ( k s ) = E 1 E 2 E 3 2 χ ( 3 ) ( - ω s , ω 1 , ω 2 , ω 3 ) 2 .
H = H S + H SB ,
H S = ν = a , b , c , d ν ( ν - i 2 γ ν ) ν ,
H S B = ν = a , b , c , d ν H SB ν ( Q B ) ν .
V = a , b , c , d ( μ a b a b + μ a d a d + μ c b c b + μ c d c d + H . c . ) ,
ρ ( - ) = a a P ( a ) a ,
ρ ( - ) = a P ( a ) a a ,
P ( a ) = exp ( - β a ) / a exp ( - β a ) .
ν λ G ( t ) ν λ s = ν λ G ( t ) ν λ s δ ν ν δ λ λ ,
ν λ G ( t ) ν λ s = exp [ - i ω ν λ t - ( 1 / 2 ) ( γ ν + γ λ ) t ] × ν λ exp ( - i L SB t ) ν λ s ,             ν , λ = a , b , c , d ,
ω ν λ ν - λ .
L SB A [ H SB , A ] .
d ρ ¯ ν λ d t = ( - i ω ν λ - Γ ν λ ) ρ ¯ ν λ ,             ν , λ = a , b , c , d ,
ρ ¯ Tr B ( ρ )
Γ ν λ = ½ ( γ ν + γ λ ) + Γ ^ ν λ .
ρ ¯ ν λ ( t ) = ν λ G ( t ) ν λ ρ ¯ ν λ ( 0 ) ,
ν λ G ( t ) ν λ Tr B ν λ G ( t ) ν λ s .
I ν λ ( t ) ν λ G ( t ) ν λ = exp [ ( i ω ν λ - Γ ν λ ) t ] .
I ^ ν λ ( ω ) ν λ G ^ ( ω ) ν λ = - i 0 d τ I ν λ ( τ ) exp ( i ω τ ) .
I ^ ν λ ( ω ) = 1 ω - ω ν λ + i Γ ν λ .
ν λ V ν λ = V ν ν δ λ λ - V * λ λ δ ν ν .
R ( t 3 , t 2 , t 1 ) = a , b , c , d P ( a ) μ a b μ b c μ c d μ d a [ - I a d ( t 3 ) I a c ( t 2 ) I a b ( t 1 ) + I d c ( t 3 ) I d b ( t 2 ) I d a ( t 1 ) + I d c ( t 3 ) I d b ( t 2 ) I a b ( t 1 ) + I d c ( t 3 ) I a c ( t 2 ) I a b ( t 1 ) + I b a ( t 3 ) I c a ( t 2 ) I d a ( t 1 ) - I c b ( t 3 ) I d b ( t 2 ) I a b ( t 1 ) - I c b ( t 3 ) I d b ( t 2 ) I d a ( t 1 ) - I c b ( t 3 ) I c a ( t ) I d a ( t 1 ) ] ,
R ^ ( ω 1 + ω 2 + ω 3 , ω 1 + ω 2 , ω 1 ) = a , b , c , d P ( a ) μ a b μ b c μ c d μ d a × [ - I ^ a d ( ω 1 + ω 2 + ω 3 ) I ^ a c ( ω 1 + ω 2 ) I ^ a b ( ω 1 ) + I ^ d c ( ω 1 + ω 2 + ω 3 ) I ^ d b ( ω 1 + ω 2 ) I ^ d a ( ω 1 ) + I ^ d c ( ω 1 + ω 2 + ω 3 ) I ^ d b ( ω 1 + ω 2 ) I ^ a b ( ω 1 ) + I ^ d c ( ω 1 + ω 2 + ω 3 ) I ^ a c ( ω 1 + ω 2 ) I ^ a b ( ω 1 ) + I ^ b a ( ω 1 + ω 2 + ω 3 ) I ^ c a ( ω 1 + ω 2 ) I ^ d a ( ω 1 ) - I ^ c b ( ω 1 + ω 2 + ω 3 ) I ^ d b ( ω 1 + ω 2 ) I ^ a b ( ω 1 ) - I ^ c b ( ω 1 + ω 2 + ω 3 ) I ^ d b ( ω 1 + ω 2 ) I ^ d a ( ω 1 ) - I ^ c b ( ω 1 + ω 2 + ω 3 ) I ^ c a ( ω 1 + ω 2 ) I ^ d a ( ω 1 ) ] .
H = ν = a , b , c , d ν [ ν - ( i / 2 ) γ ν + Δ ν ( t ) ] ν ,
Δ ν ( t ) = 0 ,             ν = a , c ,
Δ ν ( t ) = Δ ( t ) ,             ν = b , d , .
Δ ( t ) = 0 ,
Δ ( t 1 ) Δ ( t 2 ) = D 2 exp ( - Λ t 1 - t 2 ) .
R ( t 3 , t 2 , t 1 ) = V G ˜ ( t 1 + t 2 + t 3 , t 1 + t 2 ) × V G ˜ ( t 1 + t 2 , t 1 ) V G ˜ ( t 1 , 0 ) V ρ ( - ) ,
ν λ G ˜ ( τ 2 , τ 1 ) ν λ s = exp { - i ω ν λ ( τ 2 - τ 1 ) - ½ ( γ ν + γ λ ) ( τ 2 - τ 1 ) - i τ 1 τ 2 d τ [ Δ ν ( τ ) - Δ λ ( τ ) ] }
ν λ G ˜ ( τ 2 , τ 1 ) ν λ s = δ ν ν δ λ λ ν λ G ˜ ( τ 2 , τ 1 ) ν λ s .
R ( t 3 , t 2 , t 1 ) = a , b , c , d P ( a ) μ a b μ b c μ c d μ d a × [ - K a d ( t 3 ) K a c ( t 2 ) K a b ( t 1 ) Φ - ( t 3 , t 2 , t 1 ) + K d c ( t 3 ) K d b ( t 2 ) K d a ( t 1 ) Φ - ( t 3 , t 2 , t 1 ) + K d c ( t 3 ) K d b ( t 2 ) K a b ( t 1 ) Φ + ( t 3 , t 2 , t 1 ) + K d c ( t 3 ) K a c ( t 2 ) K a b ( t 1 ) Φ + ( t 3 , t 2 , t 1 ) + K b a ( t 3 ) K c a ( t 2 ) K d a ( t 1 ) Φ - ( t 3 , t 2 , t 1 ) - K c b ( t 3 ) K d b ( t 2 ) K a b ( t 1 ) Φ - ( t 3 , t 2 , t 1 ) - K c b ( t 3 ) K d b ( t 2 ) K d a ( t 1 ) Φ + ( t 3 , t 2 , t 1 ) - K c b ( t 3 ) K c a ( t 2 ) K d a ( t 1 ) Φ + ( t 3 , t 2 , t 1 ) ] ,
K ν λ ( t ) = exp [ - i ω ν λ t - ½ ( γ ν + γ λ ) t ] ,
Φ ± ( t 3 , t 2 , t 1 ) = exp { - g ( t 3 ) - g ( t 1 ) ± [ g ( t 1 + t 2 + t 3 ) + g ( t 2 ) - g ( t 1 + t 2 ) - g ( t 2 + t 3 ) ] } ,
g ( t ) = 0 t d τ 1 0 τ 1 d τ 2 Δ ( τ 1 ) Δ ( τ 2 ) = D 2 Λ 2 [ exp ( - Λ t ) - 1 + Λ t ] .
Φ ( ± ) ( t 3 , t 2 , t 1 ) = exp [ - i t 1 + t 2 t 1 + t 2 + t 3 d τ Δ ( τ ) ± i 0 t 1 d τ Δ ( τ ) ] ,
Φ ± ( t 3 , t 2 , t 1 ) = exp [ - Γ ^ ( t 3 + t 1 ) ] ,
Γ ^ = D 2 / Λ .
I ν λ ( t ) = exp ( - Γ ^ t ) K ν λ ( t ) .
Φ ± ( t 3 , t 2 , t 1 ) = exp [ - ( D 2 / 2 ) ( t 3 ± t 1 ) 2 ] .
R ^ ( ω 1 + ω 2 + ω 3 , ω 1 + ω 2 , ω 1 ) = a , b , c , d P ( a ) μ a b μ b c μ c d μ d a [ - ψ - ( s 3 + Ω a d , s 2 + Ω a c , s 1 + Ω a b ) + ψ - ( s 3 + Ω d c , s 2 + Ω d b , s 1 + Ω d a ) + ψ + ( s 3 + Ω d c , s 2 + Ω d b , s 1 + Ω a b ) + ψ + ( s 3 + Ω d c , s 2 + Ω a c , s 1 + Ω a b ) + ψ - ( s 3 + Ω b a , s 2 + Ω c a , + s 1 + Ω d a ) - ψ - ( s 3 + Ω c b , s 2 + Ω d b , s 1 + Ω a b ) - ψ + ( s 3 + Ω c b , s 2 + Ω d b , s 2 + Ω d a ) - ψ + ( s 3 + Ω c b , s 2 + Ω c a , s 1 + Ω d a ) ] ,
s 1 = - i ω 1 ,
s 2 = - i ( ω 1 + ω 2 ) ,
s 3 = - i ( ω 1 + ω 2 + ω 3 ) ,
Ω ν λ = i ω ν λ + ( 1 / 2 ) ( γ ν + γ λ ) .
ψ ± ( s 3 , s 2 , s 1 ) = 0 d t 3 0 d t 2 0 d t 1 × exp [ - s 1 t 1 - s 2 t 2 - s 3 t 3 ] ϕ ± ( t 3 , t 2 , t 1 ) .
k s = 2 k 1 - k 2 ,
ω s = 2 ω 1 - ω 2 .
χ ( 3 ) ( - ω s , ω 1 , - ω 2 , ω 1 ) = R ^ ( 2 ω 1 - ω 2 , ω 1 - ω 2 , ω 1 ) + R ^ ( 2 ω 1 - ω 2 , ω 1 - ω 2 , - ω 2 ) + R ^ ( 2 ω 1 - ω 2 , 2 ω 1 , ω 1 ) ,
ω 1 - ω 2 = ω c a
ω 1 - ω 2 = ω d b .
χ c a ( 3 ) = b , d P ( a ) μ a b μ b c μ c d μ d a [ I ^ b a ( 2 ω 1 - ω 2 ) I ^ c a ( ω 1 - ω 2 ) - I ^ c b ( 2 ω 1 - ω 2 ) I ^ c a ( ω 1 - ω 2 ) ] × [ I ^ d a ( ω 1 ) + I ^ d a ( - ω 2 ) ] ,
χ c a ( 3 ) = b , d P ( a ) μ a b μ b c μ c d μ d a I ^ b a ( 2 ω 1 - ω 2 ) I ^ c a ( ω 1 - ω 2 ) I ^ d a ( ω 1 ) .
χ d b ( 3 ) = a , c P ( a ) μ a b μ b c μ c d μ d a [ I ^ d c ( 2 ω 1 - ω 2 ) I ^ d b ( ω 1 - ω 2 ) - I ^ c b ( 2 ω 1 - ω 2 ) I ^ d b ( ω 1 - ω 2 ) ] × [ I ^ d a ( ω 1 ) + I ^ d a ( - ω 2 ) + I ^ a b ( ω 1 ) + I ^ a b ( - ω 2 ) ] .
χ d b ( 3 ) = a , c P ( a ) μ a b μ b c μ c d μ d a × I ^ d c ( 2 ω 1 - ω 2 ) I ^ d b ( ω 1 - ω 2 ) [ I ^ d a ( ω 1 ) + I ^ a b ( - ω 2 ) ] .
χ c a ( 3 ) = b , d P ( a ) μ a b μ b c μ c d μ d a 1 2 ω 1 - ω 2 - ω b a + i Γ b a × 1 ω 1 - ω 2 - ω c a + i Γ c a 1 ω 1 - ω d a + i Γ d a
χ d b ( 3 ) = a , c P ( a ) μ a b μ b c μ c d μ d a × 1 2 ω 1 - ω 2 - ω d c + i Γ d c 1 ω 1 - ω 2 - ω d b + i Γ d b × [ 1 ω 1 - ω d a + i Γ d a + 1 - ω 2 - ω a b + i Γ a b ] .
χ d b ( 3 ) = a , c P ( a ) μ a b μ b c μ c d μ d a × 1 2 ω 1 - ω 2 - ω d c + i Γ d c 1 ω 1 - ω d a + i Γ d a × 1 - ω 2 - ω a b + i Γ a b [ 1 + i Γ ˜ ω 1 - ω 2 - ω d b + i Γ d b ] ,
Γ ˜ Γ a b + Γ a d - Γ b d = γ a + Γ ^ a b + Γ ^ a d - Γ ^ b d .
P ( k s , t ) = ( - i ) 3 0 d t 3 0 d t 2 0 d t 1 R ( t 3 , t 2 , t 1 ) × E ( t - t 3 - T ) exp [ i ( 2 ω 1 - ω 2 ) t 3 ] { E * ( t - t 2 - t 3 ) × E ( t - t 1 - t 2 - t 3 ) exp [ i ω 1 t 1 + i ( ω 1 - ω 2 ) t 2 ] + E ( t - t 2 - t 3 ) E * ( t - t 1 - t 2 - t 3 ) × exp [ - i ω 2 t 1 + i ( ω 1 - ω 2 ) t 2 ] } .
E ( τ ) = E δ ( τ )
P c a ( k s , t ) = E 3 a , b , c , d P ( a ) μ a b μ b c μ c d μ d a I b a ( t - T ) I c a ( T ) × exp [ i ( 2 ω 1 - ω 2 ) t - i ω 1 T ]
P d b ( k s , t ) = 2 E 3 a , b , c , d P ( a ) μ a b μ b c μ c d μ d a I d c ( t - T ) I d b ( T ) × exp [ i ( 2 ω 1 - ω 2 ) t - i ω 1 T ] .
S c a ( T ) I c a ( T ) 2 = exp ( - 2 Γ c a T )
S d b ( T ) I d b ( T ) 2 = exp ( - 2 Γ d b T ) .
F ( τ 1 , τ 2 , τ 3 , τ 4 ) Tr [ V ( τ 1 ) V ( τ 2 ) V ( τ 3 ) V ( τ 4 ) ρ ( - ) ] = V ( τ 1 ) V ( τ 2 ) V ( τ 3 ) V ( τ 4 ) ,
V ( τ ) = exp ( i H τ ) V exp ( - i H τ ) .
R ( t 3 , t 2 , t 1 ) = - F ( 0 , t 1 , t 1 + t 2 , t 1 + t 2 + t 3 ) + F ( t 1 , t 1 + t 2 , t 1 + t 2 + t 3 , 0 ) + F ( 0 , t 1 + t 2 , t 1 + t 2 + t 3 , t 1 ) + F ( 0 , t 1 , t 1 + t 2 + t 3 , t 1 + t 2 ) + F ( t 1 + t 2 + t 3 , t 1 + t 2 , t 1 , 0 ) - F ( 0 , t 1 + t 2 + t 3 , t 1 + t 2 , t 1 ) - F ( t 1 , t 1 + t 2 + t 3 , t 1 + t 2 , 0 ) - F ( t 1 + t 2 , t 1 + t 2 + t 3 , t 1 , 0 ) ,
Φ ± ( τ 1 , τ 2 , τ 3 ) = exp { - g ( τ 1 ) - g ( τ 3 ) ± η exp ( - Λ τ 2 ) [ 1 - exp ( - Λ τ 1 ) ] [ 1 - exp ( - Λ τ 3 ) ] } ,
η = D 2 / Λ 2 .
ψ ± ( s 1 , s 2 , s 3 ) = n = 0 ( ± 1 ) n η n n ! 1 s 2 + n Λ J ˜ n ( s 1 ) J ˜ n ( s 3 ) ,
J ˜ n ( s ) = 0 d τ [ 1 - exp ( - Λ τ ) ] n exp [ - s τ - g ( τ ) ] .
η Λ J ˜ n + 1 ( s ) = n Λ J ˜ n - 1 ( s ) - ( s + n Λ ) J ˜ n ( s ) ,             n = 1 , 2 , ,
η Λ J ˜ 1 ( s ) = - J ˜ 0 ( s ) .
J ˜ 0 ( s ) = 1 s + D 2 s + Λ + 2 D 2 s + 2 Λ + 3 D 2 s + 3 Λ + .

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