It is important to recognize the trace over quantum field operators implied in these atomic-operator equations; missing here are terms representing field fluctuations needed, in a fully consistent treatment, to preserve both field and atom commutation relations. See P. L. Knight and P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980).

[CrossRef]

This particular definition of time-dependent spectra has been advocated by J. H. Eberly and K. Wodkiewicz, “The time-dependent physical spectrum of light,” J. Opt. Soc. Am. 67, 1252–1261 (1977). See also B. Renaud, R. M. Whitley, and C. R. Stroud, “Nonstationary two-level resonance fluorescence,” J. Phys. B 10, 19–35 (1977); E. Courtens and A. Szöke, “Time and spectral resolution in resonance scattering and resonance fluorescence,” Phys. Rev. A 15, 1588–1603 (1977), erratum 17, 2119 (1978); J. H. Eberly, C. V. Kunasz, and K. Wodkiewicz, “Time-dependent spectrum of resonance fluorescence,” J. Phys. B 13, 217–240 (1980).

[CrossRef]

For reviews see S. Feneuille, “Interaction of laser radiation with free atoms,” Rep. Prog. Phys. 40, 1257–1304 (1977); S. Swain, “Theory of atomic processes in strong resonant magnetic fields,” Adv. At. Mol. Phys. 16, 159–197 (1980); P. O. Knight and P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980). Recent work includes M. Le Berre-Rousseau, E. Ressayre, and A. Tallet, “Statistical properties of the light scattered by a two-level atom driven by a resonant multimode laser beam,” Phys. Rev. A 23, 2580–2593 (1981); G. Nienhuis and F. Schuller, “Collision broadening of saturated resonance fluorescence spectra by two-state atoms irradiated by monochromatic light,” J. Phys. B 13, 2205–2216 (1980); A. T. Georges and S. N. Dixit, “Laser line-shape effects in resonance fluorescence,” Phys. Rev. A 23, 2580–2593 (1981); P. Zoller, G. Alber, and R. Savador, “AC Start splitting in intense stochastic driving fields with Gaussian statistics and non-Lorentzian line shape,” Phys. Rev. A 24, 398–410 (1981); A. T. Georges, “Intensity correlations in resonance fluorescence excited by intense incoherent fields,” Opt. Commun. 38, 274–278 (1981); P. D. Kleiber, J. Cooper, K. Burnett, C. V. Kunasz, and M. G. Raymer, “Theory of time-dependent intense-field collisional resonance fluorescence,” Phys. Rev. A 27, 291–301 (1983); W. Vogel, D. G. Welsch, and K. Wodkiewicz, “Theory of resonance fluorescence in a fluctuating field,” Phys. Rev. A 28, 1546–1559 (1983).

[CrossRef]

Considerable literature treats these Heisenberg operators with attention to subtleties involving quantum behavior of field operators; see J. R. Ackerhalt and J. H. Eberly, “Quantum electrodynamics and radiation reaction: nonrelativistic atomic frequency shifts and lifetimes,” Phys. Rev. D 10, 3350–3375 (1974); K. Wodkiewicz and J. H. Eberly, “Markovian and non-Markovian behavior in two-level atom fluorescence,” Ann. Phys. 101, 574–593 (1976); J. H. Eberly, “The Heisenberg picture formalism in the theories of atom-radiation interactions,” in Laser Physics, Proceedings of the Second New Zealand Summer School in Laser Physics, D. F. Walls and J. D. Harvey, eds. (Academic, Sydney, Australia, 1980), pp. 1–31; P. Milonni, “Semiclassical and quantum-electrodynamical approaches in nonrelativistic radiation theory,” Phys. Rep. 25c, 1–81 (1976); J. R. Ackerhalt, P. L. Knight, and J. H. Eberly, “Radiation reaction and radiative frequency shifts,” Phys. Rev. Lett. 30, 456–460 (1973); P. L. Knight and P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980); B. Renaud, R. M. Whitley, and C. R. Stroud, “Correlation functions and the ac Stark effect,” J. Phys. B 9, L19–L24 (1976); P. L. Knight, W. A. Molander, and C. R. Stroud, “Asymmetric resonance fluorescence spectra in partially coherent fields,” Phys. Rev. A 17, 1547–1549. Although the equations as I use them treat atomic operators, quantized field operators are replaced by expectation values. Thus the equations yield reduced density matrices for the atom and the field appears as classical. See also D. T. Pegg, “Atomic spectroscopy: interaction of atoms with coherent fields,” in Laser Physics, Proceedings of the Second New Summer School in Laser Physics, D. F. Walls and J. D. Harvey, eds. (Academic, Sydney, Australia, 1980), pp. 33–61.

[CrossRef]

B. R. Mollow, “Stimulated emission and absorption near resonance for driven systems,” Phys. Rev. A 5, 2217–2222 (1972); A. M. Bonch-Bruevich, S. G. Prizibelskii, V. A. Khodovoi, and N. Chigir, “Investigation of the absorption spectrum of a two-level system in intense monochromatic radiation fields,” Sov. Phys. JETP 43, 230–236 (1976); G. S. Agarwal, “Quantum statistical theory of optical-resonance phenomena in fluctuating laser fields,” Phys. Rev. A 19, 1490–1506 (1978).

[CrossRef]

See particularly R. F. Fox, “Contribution to the theory of multiplicative stochastic processes,” J. Math. Phys. 13, 1196–1207 (1972); G. S. Agarwal, “Exact solution for the influence of laser temporal fluctuations on resonance fluorescence,” Phys. Rev. Lett. 37, 1383–1386 (1976); G. S. Agarwal, “Quantum statistical theory of optical-resonance phenomena in fluctuating laser fields,” Phys. Rev. A 18, 1490–1506 (1978); G. S. Agarwal and P. A. Naragana, “Effect of probe field strength and the fluctuations of the exciting laser on the asymmetry of Autler–Townes doublet,” Opt. Commun. 30, 364–368 (1979); K. Wodkiewicz, “Stochastic incoherence of optical Bloch equations,” Phys. Rev. A 19, 1686–1696 (1979); K. Wodkiewicz, “Exact solutions of some multiplicative stochastic processes,” J. Math. Phys. 20, 45–48 (1979).

[CrossRef]

C. Cohen-Tannoudji and S. Haroche, “Absorption et diffusion de photons optiques par un atome en interaction avec des photons de radiofrequence,” J. Phys. (Paris) 30, 153–173 (1969); R. M. Whitley and C. R. Stroud, “Double optical resonance,” Phys. Rev. A 14, 1498–1513 (1976); C. Cohen-Tannoudji and S. Reynaud, “Dressed-atom description of resonance fluorescence and absorption spectra of a multi-level atom in an intense laser beam,” J. Phys. B 10, 345–364, 365–383 (1976); P. R. Berman and R. Salomaa, “Comparison between dressed-atom and bare-atom pictures in laser spectroscopy,” Phys. Rev. A 25, 2667–2692 (1982).

[CrossRef]

An equivalent equation, integral in time, is used by M. Blume, ”Stochastic theory of line shape: generalizaation of the Kubo–Anderson model,” Phys. Rev. 174, 351–358 (1968) and A. Brissaud and U. Frisch, “Solving linear stochastic differential equations,” J. Math. Phys. 15, 524–534 (1974).

[CrossRef]

See A. I. Burshtein, “Kinetics of the relaxation induced by a sudden potential change,” Sov. Phys. JETP 22, 939–947 (1966); A. I. Burshtein, “Kinetics of induced relaxation,” Sov. Phys. JETP 21, 567–573 (1965); N. G. Van Kampen, “Stochastic differential equations,” Phys. Rep. 24, 171–228 (1976); M. Blume, “Stochastic theory of line shape: generalization of the Kubo–Anderson model,” Phys. Rev. 174, 351–358 (1968); P. Zoller and F. Ehlotzky, “Resonance fluorescence in modulated laser fields,” J. Phys. B 10, 3023–3032 (1977); R. Kubo, “A stochastic theory of line shape and relaxation,” in Fluctuations, Relaxations, and Resonance in Magnetic Systems, D. Ter Haar, ed. (Oliver and Boyd, Edinburgh, 1962); N. Van Kampen, “Stochastic differential equations,” Phys. Rep. 24, 171–228 (1976).

[CrossRef]

A. Burshtein, “Kinetics of the relaxation induced by a sudden potential change,” Sov. Phys. JETP 22, 939–947 (1966); A. I. Burshtein, “Kinetics of induced relaxation,” Sov. Phys. JETP 21, 567–573 (1965); A. I. Burshtein and Yu. S. Oseledchik, “Relaxation in a system subjected to suddenly changing perturbations in the presence of correlation between successive values of the perturbation,” Sov. Phys. JETP 24, 716–724 (1967); Z. D. Zusman and A. I. Burshtein, “Stark effect in the field of incoherent radiation,” Sov. Phys. JETP 34, 520–526 (1972).

M. Lax, “Formal theory of quantum fluctuations from a driven state,” Phys. Rev. 129, 2342–2348 (1963). See also B. R. Mollow, “Power spectrum of light scattered by a two-level system,” Phys. Rev. 188, 1969–1975 (1969); G. S. Agarwal, “Exact solution for the influence of laser temporal fluctuations on resonance fluorescence,” Phys. Rev. Lett. 37, 1383–1386 (1976); G. S. Agarwal, “Quantum statistical theory of optical-resonance phenomena in fluctuating laser fields,” Phys. Rev. A 18, 1490–1506 (1978); G. Nienhuis and F. Schuller, “Collision broadening of saturated resonance fluorescence spectra by two-state atoms irradiated by monochromatic light,” J. Phys. B 13, 2205–2216 (1980).

[CrossRef]

Early references include S. G. Rautian and I. I. Sobel’man, “Line shape and dispersion in the vicinity of an absorption band, as affected by induced transitions,” Sov. Phys. JETP 14, 328–333 (1962);M. Newstein, “Spontaneous emission in the presence of a prescribed classical field,” Phys. Rev. 167, 89–96 (1968); B. R. Mollow, “Power spectrum of light scattered by a two-level system,” Phys. Rev. 188, 1969–1975 (1969); L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

[CrossRef]

See R. Kubo, “Note on the stochastic theory of resonance absorption,” J. Phys. Soc. Jpn. 9, 935–944 (1954); P. W. Anderson, “A mathematical model for the narrowing of spectral lines by exchange or motion,” J. Phys. Soc. Jpn. 9, 316–339 (1954); P. Avan and C. Cohen-Tannoudji, “Two-level atom saturated by a fluctuating resonant laser beam. Calculation of the fluorescence spectrum,” J. Phys. B, 10, 155–170 (1977).

[CrossRef]

Considerable literature treats these Heisenberg operators with attention to subtleties involving quantum behavior of field operators; see J. R. Ackerhalt and J. H. Eberly, “Quantum electrodynamics and radiation reaction: nonrelativistic atomic frequency shifts and lifetimes,” Phys. Rev. D 10, 3350–3375 (1974); K. Wodkiewicz and J. H. Eberly, “Markovian and non-Markovian behavior in two-level atom fluorescence,” Ann. Phys. 101, 574–593 (1976); J. H. Eberly, “The Heisenberg picture formalism in the theories of atom-radiation interactions,” in Laser Physics, Proceedings of the Second New Zealand Summer School in Laser Physics, D. F. Walls and J. D. Harvey, eds. (Academic, Sydney, Australia, 1980), pp. 1–31; P. Milonni, “Semiclassical and quantum-electrodynamical approaches in nonrelativistic radiation theory,” Phys. Rep. 25c, 1–81 (1976); J. R. Ackerhalt, P. L. Knight, and J. H. Eberly, “Radiation reaction and radiative frequency shifts,” Phys. Rev. Lett. 30, 456–460 (1973); P. L. Knight and P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980); B. Renaud, R. M. Whitley, and C. R. Stroud, “Correlation functions and the ac Stark effect,” J. Phys. B 9, L19–L24 (1976); P. L. Knight, W. A. Molander, and C. R. Stroud, “Asymmetric resonance fluorescence spectra in partially coherent fields,” Phys. Rev. A 17, 1547–1549. Although the equations as I use them treat atomic operators, quantized field operators are replaced by expectation values. Thus the equations yield reduced density matrices for the atom and the field appears as classical. See also D. T. Pegg, “Atomic spectroscopy: interaction of atoms with coherent fields,” in Laser Physics, Proceedings of the Second New Summer School in Laser Physics, D. F. Walls and J. D. Harvey, eds. (Academic, Sydney, Australia, 1980), pp. 33–61.

[CrossRef]

An equivalent equation, integral in time, is used by M. Blume, ”Stochastic theory of line shape: generalizaation of the Kubo–Anderson model,” Phys. Rev. 174, 351–358 (1968) and A. Brissaud and U. Frisch, “Solving linear stochastic differential equations,” J. Math. Phys. 15, 524–534 (1974).

[CrossRef]

Representative references include R. G. Breene, The Shift and Shape of Spectral Lines (Pergamon, New York, 1961); H. R. Griem, Spectral Line Broadening by Plasmas (Academic, New York, 1974); F. Schuller and W. Behmenburg, “Perturbation of spectral lines by atomic interactions,” Phys. Rep. 12, 273–334 (1974); A. Ben-Reuven, “The meaning of collision broadening of spectral lines: the classical oscillator analog,” Adv. Chem. Phys. 33, 201–235 (1975); B. Wende, ed., Spectral Line Shapes (de Gruyter, Berlin, 1981).

[CrossRef]

A. Burshtein, “Kinetics of the relaxation induced by a sudden potential change,” Sov. Phys. JETP 22, 939–947 (1966); A. I. Burshtein, “Kinetics of induced relaxation,” Sov. Phys. JETP 21, 567–573 (1965); A. I. Burshtein and Yu. S. Oseledchik, “Relaxation in a system subjected to suddenly changing perturbations in the presence of correlation between successive values of the perturbation,” Sov. Phys. JETP 24, 716–724 (1967); Z. D. Zusman and A. I. Burshtein, “Stark effect in the field of incoherent radiation,” Sov. Phys. JETP 34, 520–526 (1972).

See A. I. Burshtein, “Kinetics of the relaxation induced by a sudden potential change,” Sov. Phys. JETP 22, 939–947 (1966); A. I. Burshtein, “Kinetics of induced relaxation,” Sov. Phys. JETP 21, 567–573 (1965); N. G. Van Kampen, “Stochastic differential equations,” Phys. Rep. 24, 171–228 (1976); M. Blume, “Stochastic theory of line shape: generalization of the Kubo–Anderson model,” Phys. Rev. 174, 351–358 (1968); P. Zoller and F. Ehlotzky, “Resonance fluorescence in modulated laser fields,” J. Phys. B 10, 3023–3032 (1977); R. Kubo, “A stochastic theory of line shape and relaxation,” in Fluctuations, Relaxations, and Resonance in Magnetic Systems, D. Ter Haar, ed. (Oliver and Boyd, Edinburgh, 1962); N. Van Kampen, “Stochastic differential equations,” Phys. Rep. 24, 171–228 (1976).

[CrossRef]

This behavior has been stressed by Z. D. Zusman and A. I. Burshtein, “Stark effect in the field of incoherent radiation,” Sov. Phys. JETP34, 520–526 (1972) and will be discussed in detail in papers submitted for publication.9

C. Cohen-Tannoudji and S. Haroche, “Absorption et diffusion de photons optiques par un atome en interaction avec des photons de radiofrequence,” J. Phys. (Paris) 30, 153–173 (1969); R. M. Whitley and C. R. Stroud, “Double optical resonance,” Phys. Rev. A 14, 1498–1513 (1976); C. Cohen-Tannoudji and S. Reynaud, “Dressed-atom description of resonance fluorescence and absorption spectra of a multi-level atom in an intense laser beam,” J. Phys. B 10, 345–364, 365–383 (1976); P. R. Berman and R. Salomaa, “Comparison between dressed-atom and bare-atom pictures in laser spectroscopy,” Phys. Rev. A 25, 2667–2692 (1982).

[CrossRef]

The method underlies the so-called model microfield method for treating spectral line shapes (weakly excited). See J. W. Dufty, “The microfield formulation of spectral line broadening,” in Spectral Line Shapes, B. Wende, ed. (de Gruyter, Berlin, 1981), pp. 41–62.

This particular definition of time-dependent spectra has been advocated by J. H. Eberly and K. Wodkiewicz, “The time-dependent physical spectrum of light,” J. Opt. Soc. Am. 67, 1252–1261 (1977). See also B. Renaud, R. M. Whitley, and C. R. Stroud, “Nonstationary two-level resonance fluorescence,” J. Phys. B 10, 19–35 (1977); E. Courtens and A. Szöke, “Time and spectral resolution in resonance scattering and resonance fluorescence,” Phys. Rev. A 15, 1588–1603 (1977), erratum 17, 2119 (1978); J. H. Eberly, C. V. Kunasz, and K. Wodkiewicz, “Time-dependent spectrum of resonance fluorescence,” J. Phys. B 13, 217–240 (1980).

[CrossRef]

Considerable literature treats these Heisenberg operators with attention to subtleties involving quantum behavior of field operators; see J. R. Ackerhalt and J. H. Eberly, “Quantum electrodynamics and radiation reaction: nonrelativistic atomic frequency shifts and lifetimes,” Phys. Rev. D 10, 3350–3375 (1974); K. Wodkiewicz and J. H. Eberly, “Markovian and non-Markovian behavior in two-level atom fluorescence,” Ann. Phys. 101, 574–593 (1976); J. H. Eberly, “The Heisenberg picture formalism in the theories of atom-radiation interactions,” in Laser Physics, Proceedings of the Second New Zealand Summer School in Laser Physics, D. F. Walls and J. D. Harvey, eds. (Academic, Sydney, Australia, 1980), pp. 1–31; P. Milonni, “Semiclassical and quantum-electrodynamical approaches in nonrelativistic radiation theory,” Phys. Rep. 25c, 1–81 (1976); J. R. Ackerhalt, P. L. Knight, and J. H. Eberly, “Radiation reaction and radiative frequency shifts,” Phys. Rev. Lett. 30, 456–460 (1973); P. L. Knight and P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980); B. Renaud, R. M. Whitley, and C. R. Stroud, “Correlation functions and the ac Stark effect,” J. Phys. B 9, L19–L24 (1976); P. L. Knight, W. A. Molander, and C. R. Stroud, “Asymmetric resonance fluorescence spectra in partially coherent fields,” Phys. Rev. A 17, 1547–1549. Although the equations as I use them treat atomic operators, quantized field operators are replaced by expectation values. Thus the equations yield reduced density matrices for the atom and the field appears as classical. See also D. T. Pegg, “Atomic spectroscopy: interaction of atoms with coherent fields,” in Laser Physics, Proceedings of the Second New Summer School in Laser Physics, D. F. Walls and J. D. Harvey, eds. (Academic, Sydney, Australia, 1980), pp. 33–61.

[CrossRef]

J. H. Eberly, B. W. Shore, and K. Wodkiewicz, Phys. Rev. A (to be published).

For reviews see S. Feneuille, “Interaction of laser radiation with free atoms,” Rep. Prog. Phys. 40, 1257–1304 (1977); S. Swain, “Theory of atomic processes in strong resonant magnetic fields,” Adv. At. Mol. Phys. 16, 159–197 (1980); P. O. Knight and P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980). Recent work includes M. Le Berre-Rousseau, E. Ressayre, and A. Tallet, “Statistical properties of the light scattered by a two-level atom driven by a resonant multimode laser beam,” Phys. Rev. A 23, 2580–2593 (1981); G. Nienhuis and F. Schuller, “Collision broadening of saturated resonance fluorescence spectra by two-state atoms irradiated by monochromatic light,” J. Phys. B 13, 2205–2216 (1980); A. T. Georges and S. N. Dixit, “Laser line-shape effects in resonance fluorescence,” Phys. Rev. A 23, 2580–2593 (1981); P. Zoller, G. Alber, and R. Savador, “AC Start splitting in intense stochastic driving fields with Gaussian statistics and non-Lorentzian line shape,” Phys. Rev. A 24, 398–410 (1981); A. T. Georges, “Intensity correlations in resonance fluorescence excited by intense incoherent fields,” Opt. Commun. 38, 274–278 (1981); P. D. Kleiber, J. Cooper, K. Burnett, C. V. Kunasz, and M. G. Raymer, “Theory of time-dependent intense-field collisional resonance fluorescence,” Phys. Rev. A 27, 291–301 (1983); W. Vogel, D. G. Welsch, and K. Wodkiewicz, “Theory of resonance fluorescence in a fluctuating field,” Phys. Rev. A 28, 1546–1559 (1983).

[CrossRef]

See particularly R. F. Fox, “Contribution to the theory of multiplicative stochastic processes,” J. Math. Phys. 13, 1196–1207 (1972); G. S. Agarwal, “Exact solution for the influence of laser temporal fluctuations on resonance fluorescence,” Phys. Rev. Lett. 37, 1383–1386 (1976); G. S. Agarwal, “Quantum statistical theory of optical-resonance phenomena in fluctuating laser fields,” Phys. Rev. A 18, 1490–1506 (1978); G. S. Agarwal and P. A. Naragana, “Effect of probe field strength and the fluctuations of the exciting laser on the asymmetry of Autler–Townes doublet,” Opt. Commun. 30, 364–368 (1979); K. Wodkiewicz, “Stochastic incoherence of optical Bloch equations,” Phys. Rev. A 19, 1686–1696 (1979); K. Wodkiewicz, “Exact solutions of some multiplicative stochastic processes,” J. Math. Phys. 20, 45–48 (1979).

[CrossRef]

C. Cohen-Tannoudji and S. Haroche, “Absorption et diffusion de photons optiques par un atome en interaction avec des photons de radiofrequence,” J. Phys. (Paris) 30, 153–173 (1969); R. M. Whitley and C. R. Stroud, “Double optical resonance,” Phys. Rev. A 14, 1498–1513 (1976); C. Cohen-Tannoudji and S. Reynaud, “Dressed-atom description of resonance fluorescence and absorption spectra of a multi-level atom in an intense laser beam,” J. Phys. B 10, 345–364, 365–383 (1976); P. R. Berman and R. Salomaa, “Comparison between dressed-atom and bare-atom pictures in laser spectroscopy,” Phys. Rev. A 25, 2667–2692 (1982).

[CrossRef]

It is important to recognize the trace over quantum field operators implied in these atomic-operator equations; missing here are terms representing field fluctuations needed, in a fully consistent treatment, to preserve both field and atom commutation relations. See P. L. Knight and P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980).

[CrossRef]

See R. Kubo, “Note on the stochastic theory of resonance absorption,” J. Phys. Soc. Jpn. 9, 935–944 (1954); P. W. Anderson, “A mathematical model for the narrowing of spectral lines by exchange or motion,” J. Phys. Soc. Jpn. 9, 316–339 (1954); P. Avan and C. Cohen-Tannoudji, “Two-level atom saturated by a fluctuating resonant laser beam. Calculation of the fluorescence spectrum,” J. Phys. B, 10, 155–170 (1977).

[CrossRef]

The Burshtein method, which deals with Heisenberg equations, is equivalent to the Liouville equation approach used for weak-field excitation and is sometimes called the Kubo–Anderson master equation (see Ref. 6); R. Kubo, “A stochastic theory of line shape and relaxation,” in Fluctuations, Relaxations, and Resonance in Magnetic Systems, D. Ter Haar, ed. (Oliver and Boyd, Edinburgh, 1962), pp. 23–68; see M. Blume, “Stochastic theory of line shape: generalization of the Kubo–Anderson model,” Phys. Rev. 174, 351–358 (1968); U. Frisch and A. Brissaud, “Theory of Stark broadening. II. Exact line profile with model microfield,” J. Quant. Spectrosc. Radiat. Transfer 11, 1767–1783 (1971); A. Brissaud and U. Frisch, “Solving linear stochastic differential equations,” J. Math. Phys. 15, 524–534 (1974); J. Seidel, “Hydrogen Stark broadening by model electronic microfields,” Z. Naturforsch. 32a, 1195–1206 (1977). The Burshtein equation treated here can be recognized as a special case of the single- and two-time master equations derived by G. S. Agarwal, “Master equations for time correlation functions of a quantum system interacting with stochastic perturbations and applications to emission and absorption line shapes,” Z. Phys. B 33, 111–124 (1979).

[CrossRef]

M. Lax, “Formal theory of quantum fluctuations from a driven state,” Phys. Rev. 129, 2342–2348 (1963). See also B. R. Mollow, “Power spectrum of light scattered by a two-level system,” Phys. Rev. 188, 1969–1975 (1969); G. S. Agarwal, “Exact solution for the influence of laser temporal fluctuations on resonance fluorescence,” Phys. Rev. Lett. 37, 1383–1386 (1976); G. S. Agarwal, “Quantum statistical theory of optical-resonance phenomena in fluctuating laser fields,” Phys. Rev. A 18, 1490–1506 (1978); G. Nienhuis and F. Schuller, “Collision broadening of saturated resonance fluorescence spectra by two-state atoms irradiated by monochromatic light,” J. Phys. B 13, 2205–2216 (1980).

[CrossRef]

It is important to recognize the trace over quantum field operators implied in these atomic-operator equations; missing here are terms representing field fluctuations needed, in a fully consistent treatment, to preserve both field and atom commutation relations. See P. L. Knight and P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980).

[CrossRef]

B. R. Mollow, “Stimulated emission and absorption near resonance for driven systems,” Phys. Rev. A 5, 2217–2222 (1972); A. M. Bonch-Bruevich, S. G. Prizibelskii, V. A. Khodovoi, and N. Chigir, “Investigation of the absorption spectrum of a two-level system in intense monochromatic radiation fields,” Sov. Phys. JETP 43, 230–236 (1976); G. S. Agarwal, “Quantum statistical theory of optical-resonance phenomena in fluctuating laser fields,” Phys. Rev. A 19, 1490–1506 (1978).

[CrossRef]

Early references include S. G. Rautian and I. I. Sobel’man, “Line shape and dispersion in the vicinity of an absorption band, as affected by induced transitions,” Sov. Phys. JETP 14, 328–333 (1962);M. Newstein, “Spontaneous emission in the presence of a prescribed classical field,” Phys. Rev. 167, 89–96 (1968); B. R. Mollow, “Power spectrum of light scattered by a two-level system,” Phys. Rev. 188, 1969–1975 (1969); L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

[CrossRef]

Treatment of jump processes, also termed Poisson step processes, can be found in many places, including S. O. Rice, “Mathematical analysis of random noise,” in Selected Topics in Noise and Stochastic Processes, N. Wax, ed. (Dover, New York, 1954), pp. 176–181; P. W. Anderson, “A mathematical model for the narrowing of spectral lines by exchange or motion,” J. Phys. Soc. Jpn. 9, 316–339 (1954); R. Kubo, “Note on the stochastic theory of resonance absorption,” J. Phys. Soc. Jpn. 9, 935–944 (1954); R. Bourret, “Stochastic systems equivalent to second quantized systems: examples,” Can. J. Phys. 44, 2519–2524 (1966); A. Brissaud and U. Frisch, “Solving linear stochastic differential equations,” J. Math. Phys. 15, 524–534 (1974); P. Zoller and F. Ehlotzky, “Resonance fluorescence in modulated laser fields,” J. Phys. B 10, 3023–3032 (1977).

[CrossRef]

J. H. Eberly, B. W. Shore, and K. Wodkiewicz, Phys. Rev. A (to be published).

Early references include S. G. Rautian and I. I. Sobel’man, “Line shape and dispersion in the vicinity of an absorption band, as affected by induced transitions,” Sov. Phys. JETP 14, 328–333 (1962);M. Newstein, “Spontaneous emission in the presence of a prescribed classical field,” Phys. Rev. 167, 89–96 (1968); B. R. Mollow, “Power spectrum of light scattered by a two-level system,” Phys. Rev. 188, 1969–1975 (1969); L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

[CrossRef]

This particular definition of time-dependent spectra has been advocated by J. H. Eberly and K. Wodkiewicz, “The time-dependent physical spectrum of light,” J. Opt. Soc. Am. 67, 1252–1261 (1977). See also B. Renaud, R. M. Whitley, and C. R. Stroud, “Nonstationary two-level resonance fluorescence,” J. Phys. B 10, 19–35 (1977); E. Courtens and A. Szöke, “Time and spectral resolution in resonance scattering and resonance fluorescence,” Phys. Rev. A 15, 1588–1603 (1977), erratum 17, 2119 (1978); J. H. Eberly, C. V. Kunasz, and K. Wodkiewicz, “Time-dependent spectrum of resonance fluorescence,” J. Phys. B 13, 217–240 (1980).

[CrossRef]

J. H. Eberly, B. W. Shore, and K. Wodkiewicz, Phys. Rev. A (to be published).

This behavior has been stressed by Z. D. Zusman and A. I. Burshtein, “Stark effect in the field of incoherent radiation,” Sov. Phys. JETP34, 520–526 (1972) and will be discussed in detail in papers submitted for publication.9

See particularly R. F. Fox, “Contribution to the theory of multiplicative stochastic processes,” J. Math. Phys. 13, 1196–1207 (1972); G. S. Agarwal, “Exact solution for the influence of laser temporal fluctuations on resonance fluorescence,” Phys. Rev. Lett. 37, 1383–1386 (1976); G. S. Agarwal, “Quantum statistical theory of optical-resonance phenomena in fluctuating laser fields,” Phys. Rev. A 18, 1490–1506 (1978); G. S. Agarwal and P. A. Naragana, “Effect of probe field strength and the fluctuations of the exciting laser on the asymmetry of Autler–Townes doublet,” Opt. Commun. 30, 364–368 (1979); K. Wodkiewicz, “Stochastic incoherence of optical Bloch equations,” Phys. Rev. A 19, 1686–1696 (1979); K. Wodkiewicz, “Exact solutions of some multiplicative stochastic processes,” J. Math. Phys. 20, 45–48 (1979).

[CrossRef]

This particular definition of time-dependent spectra has been advocated by J. H. Eberly and K. Wodkiewicz, “The time-dependent physical spectrum of light,” J. Opt. Soc. Am. 67, 1252–1261 (1977). See also B. Renaud, R. M. Whitley, and C. R. Stroud, “Nonstationary two-level resonance fluorescence,” J. Phys. B 10, 19–35 (1977); E. Courtens and A. Szöke, “Time and spectral resolution in resonance scattering and resonance fluorescence,” Phys. Rev. A 15, 1588–1603 (1977), erratum 17, 2119 (1978); J. H. Eberly, C. V. Kunasz, and K. Wodkiewicz, “Time-dependent spectrum of resonance fluorescence,” J. Phys. B 13, 217–240 (1980).

[CrossRef]

C. Cohen-Tannoudji and S. Haroche, “Absorption et diffusion de photons optiques par un atome en interaction avec des photons de radiofrequence,” J. Phys. (Paris) 30, 153–173 (1969); R. M. Whitley and C. R. Stroud, “Double optical resonance,” Phys. Rev. A 14, 1498–1513 (1976); C. Cohen-Tannoudji and S. Reynaud, “Dressed-atom description of resonance fluorescence and absorption spectra of a multi-level atom in an intense laser beam,” J. Phys. B 10, 345–364, 365–383 (1976); P. R. Berman and R. Salomaa, “Comparison between dressed-atom and bare-atom pictures in laser spectroscopy,” Phys. Rev. A 25, 2667–2692 (1982).

[CrossRef]

See R. Kubo, “Note on the stochastic theory of resonance absorption,” J. Phys. Soc. Jpn. 9, 935–944 (1954); P. W. Anderson, “A mathematical model for the narrowing of spectral lines by exchange or motion,” J. Phys. Soc. Jpn. 9, 316–339 (1954); P. Avan and C. Cohen-Tannoudji, “Two-level atom saturated by a fluctuating resonant laser beam. Calculation of the fluorescence spectrum,” J. Phys. B, 10, 155–170 (1977).

[CrossRef]

It is important to recognize the trace over quantum field operators implied in these atomic-operator equations; missing here are terms representing field fluctuations needed, in a fully consistent treatment, to preserve both field and atom commutation relations. See P. L. Knight and P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980).

[CrossRef]

M. Lax, “Formal theory of quantum fluctuations from a driven state,” Phys. Rev. 129, 2342–2348 (1963). See also B. R. Mollow, “Power spectrum of light scattered by a two-level system,” Phys. Rev. 188, 1969–1975 (1969); G. S. Agarwal, “Exact solution for the influence of laser temporal fluctuations on resonance fluorescence,” Phys. Rev. Lett. 37, 1383–1386 (1976); G. S. Agarwal, “Quantum statistical theory of optical-resonance phenomena in fluctuating laser fields,” Phys. Rev. A 18, 1490–1506 (1978); G. Nienhuis and F. Schuller, “Collision broadening of saturated resonance fluorescence spectra by two-state atoms irradiated by monochromatic light,” J. Phys. B 13, 2205–2216 (1980).

[CrossRef]

An equivalent equation, integral in time, is used by M. Blume, ”Stochastic theory of line shape: generalizaation of the Kubo–Anderson model,” Phys. Rev. 174, 351–358 (1968) and A. Brissaud and U. Frisch, “Solving linear stochastic differential equations,” J. Math. Phys. 15, 524–534 (1974).

[CrossRef]

B. R. Mollow, “Stimulated emission and absorption near resonance for driven systems,” Phys. Rev. A 5, 2217–2222 (1972); A. M. Bonch-Bruevich, S. G. Prizibelskii, V. A. Khodovoi, and N. Chigir, “Investigation of the absorption spectrum of a two-level system in intense monochromatic radiation fields,” Sov. Phys. JETP 43, 230–236 (1976); G. S. Agarwal, “Quantum statistical theory of optical-resonance phenomena in fluctuating laser fields,” Phys. Rev. A 19, 1490–1506 (1978).

[CrossRef]

Considerable literature treats these Heisenberg operators with attention to subtleties involving quantum behavior of field operators; see J. R. Ackerhalt and J. H. Eberly, “Quantum electrodynamics and radiation reaction: nonrelativistic atomic frequency shifts and lifetimes,” Phys. Rev. D 10, 3350–3375 (1974); K. Wodkiewicz and J. H. Eberly, “Markovian and non-Markovian behavior in two-level atom fluorescence,” Ann. Phys. 101, 574–593 (1976); J. H. Eberly, “The Heisenberg picture formalism in the theories of atom-radiation interactions,” in Laser Physics, Proceedings of the Second New Zealand Summer School in Laser Physics, D. F. Walls and J. D. Harvey, eds. (Academic, Sydney, Australia, 1980), pp. 1–31; P. Milonni, “Semiclassical and quantum-electrodynamical approaches in nonrelativistic radiation theory,” Phys. Rep. 25c, 1–81 (1976); J. R. Ackerhalt, P. L. Knight, and J. H. Eberly, “Radiation reaction and radiative frequency shifts,” Phys. Rev. Lett. 30, 456–460 (1973); P. L. Knight and P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980); B. Renaud, R. M. Whitley, and C. R. Stroud, “Correlation functions and the ac Stark effect,” J. Phys. B 9, L19–L24 (1976); P. L. Knight, W. A. Molander, and C. R. Stroud, “Asymmetric resonance fluorescence spectra in partially coherent fields,” Phys. Rev. A 17, 1547–1549. Although the equations as I use them treat atomic operators, quantized field operators are replaced by expectation values. Thus the equations yield reduced density matrices for the atom and the field appears as classical. See also D. T. Pegg, “Atomic spectroscopy: interaction of atoms with coherent fields,” in Laser Physics, Proceedings of the Second New Summer School in Laser Physics, D. F. Walls and J. D. Harvey, eds. (Academic, Sydney, Australia, 1980), pp. 33–61.

[CrossRef]

For reviews see S. Feneuille, “Interaction of laser radiation with free atoms,” Rep. Prog. Phys. 40, 1257–1304 (1977); S. Swain, “Theory of atomic processes in strong resonant magnetic fields,” Adv. At. Mol. Phys. 16, 159–197 (1980); P. O. Knight and P. W. Milonni, “The Rabi frequency in optical spectra,” Phys. Rep. 66, 21–107 (1980). Recent work includes M. Le Berre-Rousseau, E. Ressayre, and A. Tallet, “Statistical properties of the light scattered by a two-level atom driven by a resonant multimode laser beam,” Phys. Rev. A 23, 2580–2593 (1981); G. Nienhuis and F. Schuller, “Collision broadening of saturated resonance fluorescence spectra by two-state atoms irradiated by monochromatic light,” J. Phys. B 13, 2205–2216 (1980); A. T. Georges and S. N. Dixit, “Laser line-shape effects in resonance fluorescence,” Phys. Rev. A 23, 2580–2593 (1981); P. Zoller, G. Alber, and R. Savador, “AC Start splitting in intense stochastic driving fields with Gaussian statistics and non-Lorentzian line shape,” Phys. Rev. A 24, 398–410 (1981); A. T. Georges, “Intensity correlations in resonance fluorescence excited by intense incoherent fields,” Opt. Commun. 38, 274–278 (1981); P. D. Kleiber, J. Cooper, K. Burnett, C. V. Kunasz, and M. G. Raymer, “Theory of time-dependent intense-field collisional resonance fluorescence,” Phys. Rev. A 27, 291–301 (1983); W. Vogel, D. G. Welsch, and K. Wodkiewicz, “Theory of resonance fluorescence in a fluctuating field,” Phys. Rev. A 28, 1546–1559 (1983).

[CrossRef]

Early references include S. G. Rautian and I. I. Sobel’man, “Line shape and dispersion in the vicinity of an absorption band, as affected by induced transitions,” Sov. Phys. JETP 14, 328–333 (1962);M. Newstein, “Spontaneous emission in the presence of a prescribed classical field,” Phys. Rev. 167, 89–96 (1968); B. R. Mollow, “Power spectrum of light scattered by a two-level system,” Phys. Rev. 188, 1969–1975 (1969); L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

[CrossRef]

A. Burshtein, “Kinetics of the relaxation induced by a sudden potential change,” Sov. Phys. JETP 22, 939–947 (1966); A. I. Burshtein, “Kinetics of induced relaxation,” Sov. Phys. JETP 21, 567–573 (1965); A. I. Burshtein and Yu. S. Oseledchik, “Relaxation in a system subjected to suddenly changing perturbations in the presence of correlation between successive values of the perturbation,” Sov. Phys. JETP 24, 716–724 (1967); Z. D. Zusman and A. I. Burshtein, “Stark effect in the field of incoherent radiation,” Sov. Phys. JETP 34, 520–526 (1972).

See A. I. Burshtein, “Kinetics of the relaxation induced by a sudden potential change,” Sov. Phys. JETP 22, 939–947 (1966); A. I. Burshtein, “Kinetics of induced relaxation,” Sov. Phys. JETP 21, 567–573 (1965); N. G. Van Kampen, “Stochastic differential equations,” Phys. Rep. 24, 171–228 (1976); M. Blume, “Stochastic theory of line shape: generalization of the Kubo–Anderson model,” Phys. Rev. 174, 351–358 (1968); P. Zoller and F. Ehlotzky, “Resonance fluorescence in modulated laser fields,” J. Phys. B 10, 3023–3032 (1977); R. Kubo, “A stochastic theory of line shape and relaxation,” in Fluctuations, Relaxations, and Resonance in Magnetic Systems, D. Ter Haar, ed. (Oliver and Boyd, Edinburgh, 1962); N. Van Kampen, “Stochastic differential equations,” Phys. Rep. 24, 171–228 (1976).

[CrossRef]

Analytic methods will be found in Ref. 5; transform methods are used in Ref. 8.

Treatment of jump processes, also termed Poisson step processes, can be found in many places, including S. O. Rice, “Mathematical analysis of random noise,” in Selected Topics in Noise and Stochastic Processes, N. Wax, ed. (Dover, New York, 1954), pp. 176–181; P. W. Anderson, “A mathematical model for the narrowing of spectral lines by exchange or motion,” J. Phys. Soc. Jpn. 9, 316–339 (1954); R. Kubo, “Note on the stochastic theory of resonance absorption,” J. Phys. Soc. Jpn. 9, 935–944 (1954); R. Bourret, “Stochastic systems equivalent to second quantized systems: examples,” Can. J. Phys. 44, 2519–2524 (1966); A. Brissaud and U. Frisch, “Solving linear stochastic differential equations,” J. Math. Phys. 15, 524–534 (1974); P. Zoller and F. Ehlotzky, “Resonance fluorescence in modulated laser fields,” J. Phys. B 10, 3023–3032 (1977).

[CrossRef]

The Burshtein method, which deals with Heisenberg equations, is equivalent to the Liouville equation approach used for weak-field excitation and is sometimes called the Kubo–Anderson master equation (see Ref. 6); R. Kubo, “A stochastic theory of line shape and relaxation,” in Fluctuations, Relaxations, and Resonance in Magnetic Systems, D. Ter Haar, ed. (Oliver and Boyd, Edinburgh, 1962), pp. 23–68; see M. Blume, “Stochastic theory of line shape: generalization of the Kubo–Anderson model,” Phys. Rev. 174, 351–358 (1968); U. Frisch and A. Brissaud, “Theory of Stark broadening. II. Exact line profile with model microfield,” J. Quant. Spectrosc. Radiat. Transfer 11, 1767–1783 (1971); A. Brissaud and U. Frisch, “Solving linear stochastic differential equations,” J. Math. Phys. 15, 524–534 (1974); J. Seidel, “Hydrogen Stark broadening by model electronic microfields,” Z. Naturforsch. 32a, 1195–1206 (1977). The Burshtein equation treated here can be recognized as a special case of the single- and two-time master equations derived by G. S. Agarwal, “Master equations for time correlation functions of a quantum system interacting with stochastic perturbations and applications to emission and absorption line shapes,” Z. Phys. B 33, 111–124 (1979).

[CrossRef]

The method underlies the so-called model microfield method for treating spectral line shapes (weakly excited). See J. W. Dufty, “The microfield formulation of spectral line broadening,” in Spectral Line Shapes, B. Wende, ed. (de Gruyter, Berlin, 1981), pp. 41–62.

J. H. Eberly, B. W. Shore, and K. Wodkiewicz, Phys. Rev. A (to be published).

Representative references include R. G. Breene, The Shift and Shape of Spectral Lines (Pergamon, New York, 1961); H. R. Griem, Spectral Line Broadening by Plasmas (Academic, New York, 1974); F. Schuller and W. Behmenburg, “Perturbation of spectral lines by atomic interactions,” Phys. Rep. 12, 273–334 (1974); A. Ben-Reuven, “The meaning of collision broadening of spectral lines: the classical oscillator analog,” Adv. Chem. Phys. 33, 201–235 (1975); B. Wende, ed., Spectral Line Shapes (de Gruyter, Berlin, 1981).

[CrossRef]

General references include N. Wax, ed., Selected Papers on Noise and Stochastic Processes (Dover, New York, 1954); M. Lax, “Fluctuations from the nonequilibrium steady state,” Rev. Mod. Phys. 32, 25–64 (1960); J. L. Doob, Stochastic Processes (Wiley, New York, 1967), Chap. V; N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, New York, 1981). Applications to differential equations include R. Kurbo, “Stochastic Liouville equations,” J. Math. Phys. 4, 174–183 (1963); N. G. Van Kampen, “Stochastic differential equations,” Phys. Rep. 24, 171–228 (1976); R. F. Fox, “Gaussian stochastic processes in physics,” Phys. Rep. 48C, 179–283 (1978); I. Oppenheim, K. E. Shuler, and G. H. Weiss, Stochastic Processes in Chemical Physics: The Master Equation (MIT, Cambridge, Mass., 1977).

[CrossRef]

This behavior has been stressed by Z. D. Zusman and A. I. Burshtein, “Stark effect in the field of incoherent radiation,” Sov. Phys. JETP34, 520–526 (1972) and will be discussed in detail in papers submitted for publication.9