Unified non-Markovian, nonsecular microscopic theory of collisional redistribution of radiation in strong resonant laser fields

When a two-level atom in a perturber gas is driven by a strong monochromatic laser, collisions transfer photons between the laser frequency and the atomic transition frequency (collisional redistribution). Reynaud and Cohen-Tannoudji (J. Physique 43, 1021, 1982) described this regime with a Markovian, secular master equation for the dressed-atom density matrix, expressing all line positions, widths, weights, and the laser-photon absorption through dressed-atom relaxation parameters that remain valid beyond the optical-Bloch domain. That framework, however, is restricted to three frequency intervals around the dressed-atom Bohr frequencies, discards non-Markovian memory on the collision timescale and nonsecular couplings (which dominate in the far wings), becomes critically Markov-limited when the generalized Rabi frequency satisfies $\Omega\tau_c\gtrsim1$, and leaves the collision spectral density $\tilde J(\omega)$ undetermined microscopically. The open problem is: a single microscopic theory that incorporates non-Markovian memory and nonsecular couplings on an equal footing, predicts the full redistribution spectrum from core intervals to far wings including the $\Omega\tau_c\gtrsim1$ regime, and determines $\tilde J(\omega)$ from first principles or a defined experimental inversion. The problem remains open.

Background

When an atom in a buffer gas is irradiated by a laser tuned near, but not on, an atomic transition, the scattered light contains more than the elastically scattered Rayleigh component at the laser frequency $\omega_L$: collisions with perturbers can supply or absorb the mismatch between $\omega_L$ and the atomic frequency $\omega_0$, so the atom can absorb a photon at $\omega_L$ and emit near $\omega_0$. This transfer of photons from the laser frequency to the atomic frequency — collisional redistribution — has been studied since the 1970s, experimentally (e.g., the saturation measurements of Carlsten, Szöke and Raymer in strontium vapor) and theoretically (the density-operator approach of Nienhuis and Schuller). The underlying bare-atom impact theory of pressure broadening (Lorentz–Anderson–Baranger line theory) treats each collision as effectively instantaneous and well separated from the next (the impact approximation), and characterizes collisions by a single damping rate $\gamma$ of the atomic dipole.

Strong resonant fields change the picture qualitatively. A two-level atom driven with Rabi frequency $\omega_1 = |\mathbf{D}\cdot\mathbf{E}|/\hbar$ (the laser–dipole coupling strength) and detuning $\delta = \omega_L - \omega_0$ undergoes coherent precession at the generalized Rabi frequency $\Omega = (\omega_1^2 + \delta^2)^{1/2}$, and its fluorescence splits into the Mollow triplet: a central component at $\omega_L$ and sidebands at $\omega_L \pm \Omega$. The standard strong-field description, the optical Bloch equations for the atomic density matrix with collision terms added, requires the whole spectrum to lie inside the impact domain, which amounts to $\Omega\tau_c \ll 1$ with $\tau_c$ the collision duration — so it fails precisely for intense fields or short collisions.

Reynaud and Cohen-Tannoudji (1982) removed this restriction by treating collisions in the dressed-atom basis. In the dressed-atom picture the laser–atom coupling is diagonalized first: the eigenstates of the coupled (atom + laser mode) system form manifolds of dressed states $\{|1,n\rangle, |2,n\rangle\}$ split by $\hbar\Omega$, and collisions then act as transitions and dephasings within this basis — a modification of the radiative cascade of the dressed atom (the sequence of fluorescence photons by which the dressed atom relaxes through its manifolds). Every characteristic of the absorption and emission spectra (positions, widths, weights of the three components, and the net absorption of laser photons) is expressed through dressed-atom relaxation parameters of $T_1$ type (the population redistribution rate $w$ between dressed states) and $T_2$ type (the transverse relaxation $\kappa$). The derivation coarse-grains the density matrix over an interval $\Delta t$ with $\tau_c \ll \Delta t \ll T_R$ ($T_R$ the relaxation time) — the Markov approximation, which discards dynamics on the collision timescale — and then drops couplings between density-matrix elements evolving at different dressed-atom Bohr frequencies (transition frequencies) $\omega_L + \alpha\Omega$ ($\alpha = 0, \pm1$) — the secular approximation, valid when the three lines are well separated ($\Omega$ large against all relaxation rates). The spectrum follows from the master equation via the quantum regression theorem (the rule that the two-time dipole correlation functions which determine the spectrum relax like the density matrix itself, valid under the same Markov conditions).

The verified pay-off is a Markovian dressed-atom description of line positions, widths, weights and integrated absorption that remains valid even for $\Omega\tau_c \gtrsim 1$, well beyond the optical-Bloch domain. The price is restricted spectral validity: the redistribution spectrum is correct only inside three frequency intervals around $\omega_L + \alpha\Omega$, subject to the paper's two separate restrictions — a Markov restriction set by the coarse-graining scale (of order $\tau_c^{-1}$) and a secular restriction set by the line separation (of order $\Omega$):

$$\big|\omega-(\omega_L+\alpha\Omega)\big|\lesssim\tau_c^{-1}\ \text{(Markov)},\qquad\big|\omega-(\omega_L+\alpha\Omega)\big|\lesssim\Omega\ \text{(secular)},\qquad\alpha=0,\pm1.$$

Outside these intervals the neglected physics is not small: non-Markovian temporal correlations of the collisional perturbation on the scale $\tau_c$, and nonsecular couplings whose dispersion-shaped (odd, derivative-like) contributions fall off more slowly in the wings than the absorption-shaped (even, peak-like) secular terms, so that in the far wings they can rival the secular ones. Moreover, for $\Omega\tau_c > 1$ the Markov approximation (not the secular one) is the critical restriction, and the authors state that in this regime it is not worthwhile to compute nonsecular corrections because the non-Markovian error dominates.

The paper also traces the relaxation parameters toward microscopic collision physics. In a spin representation (the dressed atom as a fictitious spin precessing about an effective field, the collisional perturbation acting as a fluctuating field $h(t)$), the parameters are Fourier transforms of the correlation function of $h(t)$ in the weak-collision limit (motional narrowing: many weak perturbations with $h\tau_c \ll 1$, whose rapid fluctuations average out and narrow the response); for strong collisions the structure survives through a generalized correlation function $\check C(\tau)$, so the relaxation rates sample a spectral density $\tilde J(\omega)$ at the frequencies $0$ and $\Omega$. In the far-wing weak-excitation limit, $\tilde J(\delta)$ coincides with the phenomenological parameter $\gamma(b)$ introduced by Carlsten, Szöke and Raymer to fit redistribution data, so far-wing absorption measurements probe the frequency dependence of $\tilde J(\omega)$. The paper does not determine $\tilde J(\omega)$ microscopically for any realistic collision model, and does not construct the generalized (frequency-dependent) optical Bloch equations needed in the wings, noting only the analogy with frequency-dependent relaxation matrices in magnetic resonance (Bloch, 1957).

What the 1982 paper left open is therefore sharp: a microscopic theory of collisional redistribution that keeps non-Markovian memory and nonsecular couplings on an equal footing, so that the full spectrum — core intervals and far wings, including the regime $\Omega\tau_c \gtrsim 1$ — is predicted from a specified collision model, with $\tilde J(\omega)$ computed from first principles or extracted through a defined experimental inversion. Two neighboring programs supply parts of this physics without closing it: the optical-collision (dressed-molecule) program treats the colliding pair nonperturbatively in the laser field, system by system, through close-coupled scattering calculations, and the quasi-static far-wing line-shape theory (Szudy and Baylis), in which the perturber is treated as essentially frozen during the emission, covers the wings without strong driving; neither yields a master-equation theory uniform across regimes.

Problem Statement

Construct a microscopic theory of collisional redistribution of radiation for a two-level atom driven by a strong monochromatic field and perturbed by binary elastic collisions (the model system of Reynaud and Cohen-Tannoudji, 1982) that removes the Markov and secular approximations of that treatment simultaneously and delivers the full redistribution spectrum. The theory must:

(i) retain non-Markovian memory — the temporal correlations of the collisional perturbation on the timescale of the collision duration $\tau_c$ — and nonsecular couplings — the couplings between dressed-atom density-matrix elements evolving at different Bohr frequencies $\omega_L + \alpha\Omega$ ($\alpha = 0, \pm1$) — on an equal footing, so that the redistribution spectrum is obtained uniformly from the three core intervals around $\omega_L + \alpha\Omega$ in which the 1982 relaxation-equation results hold (Markov validity of order $\tau_c^{-1}$, secular validity of order $\Omega$) out to the far wings, where the source states the dispersion-shaped nonsecular contributions may become as important as the secular ones;

(ii) remain valid in the regime $\Omega\tau_c \gtrsim 1$, where the generalized Rabi frequency is comparable to or larger than the inverse collision time and the Markov approximation, not the secular one, is the critical restriction (the 1982 treatment gives there only the main line characteristics — positions, widths, weights, integrated absorption — and nothing about the spectrum outside the core intervals);

(iii) take a specified microscopic collision model (the collision S matrix, or the correlation function $\check C(\tau)$ of the collisional perturbation, eqs. (5.7)–(5.9) of the source) as input and determine the spectral density $\tilde J(\omega)$ entering the dressed-atom relaxation parameters — defined as the Fourier transform of $\check C(\tau)$ — either by first-principles computation for explicitly specified atom-perturber systems, or through a well-defined inversion of measured far-wing absorption profiles that yields the frequency dependence of $\tilde J(\omega)$ beyond the single-parameter phenomenological fit (the source notes in section 5.1 that the deviation of far-wing absorption from the lorentzian shape gives access to the variation of $\tilde J(\delta)$ with $\delta$; section 5.2.3 identifies $\tilde J(\delta)$ in the far-wing weak-excitation limit with the parameter $\gamma(b)$ of Carlsten, Szöke and Raymer; and generalized optical Bloch equations for the wings require both $\tilde J(0)$ and $\tilde J(\delta)$);

(iv) reduce explicitly to the Markov-secular dressed-atom results of the 1982 paper inside the three core intervals, and, in the domain $\Omega\tau_c\ll1$ where both hold, to the optical Bloch equations (the reduction established in the source's section 4.3).

The theory must yield a redistribution spectrum uniformly valid from a specified microscopic collision model (a) inside the three core intervals around $\omega_L+\alpha\Omega$, (b) in the far wings where the nonsecular contributions are comparable to the secular ones, and (c) in both the $\Omega\tau_c\ll1$ and $\Omega\tau_c\gtrsim1$ regimes. The physical scope (two-level atom, monochromatic strong driving, binary elastic collisions, classical atomic motion) is that of the source paper, with quenching, degeneracy, velocity-changing and quantum-motion extensions recognized as further tasks rather than folded into this core.

References

1. S. Reynaud and C. Cohen-Tannoudji, 'Dressed atom approach to collisional redistribution', Journal de Physique 43 (7), 1021-1035 (1982). DOI: 10.1051/jphys:019820043070102100, https://doi.org/10.1051/jphys:019820043070102100
2. J. L. Carlsten, A. Szöke and M. G. Raymer, 'Collisional redistribution and saturation of near-resonance scattered light', Physical Review A 15, 1029-1045 (1977). DOI: 10.1103/physreva.15.1029, https://doi.org/10.1103/physreva.15.1029
3. G. Nienhuis and F. Schuller, 'Collisional redistribution of fluorescence radiation', Physica B+C 92, 397-408 (1977). DOI: 10.1016/0378-4363(77)90138-3, https://doi.org/10.1016/0378-4363(77)90138-3
4. F. Bloch, 'Generalized Theory of Relaxation', Physical Review 105, 1206-1222 (1957). DOI: 10.1103/physrev.105.1206, https://doi.org/10.1103/physrev.105.1206
5. G. Nienhuis, 'Correlation effects in collisional depolarisation and redistribution of radiation', Journal of Physics B: Atomic and Molecular Physics 16, 1-24 (1983). DOI: 10.1088/0022-3700/16-1-003, https://doi.org/10.1088/0022-3700/16-1-003
6. H. F. Arnoldus and G. Nienhuis, 'Collisional redistribution of intense phase-fluctuation radiation', Journal of Physics B: Atomic and Molecular Physics 16, 2325-2337 (1983). DOI: 10.1088/0022-3700/16-13-007, https://doi.org/10.1088/0022-3700/16-13-007
7. G. Nienhuis, 'Time-dependent spectra of colliding atoms in an intense radiation field', Journal of Physics B: Atomic and Molecular Physics 16, 2677-2693 (1983). DOI: 10.1088/0022-3700/16-15-011, https://doi.org/10.1088/0022-3700/16-15-011
8. F. Schuller and G. Nienhuis, 'Radiative transition rates of collisionally perturbed atoms in an intense monochromatic field', Physica B+C 121, 281-294 (1983). DOI: 10.1016/0378-4363(83)90153-5, https://doi.org/10.1016/0378-4363(83)90153-5
9. F. Schuller, 'Optical collisions and redistribution in the presence of an intense driving field', Annales de Physique 9, 609-615 (1984). DOI: 10.1051/anphys:0198400904060900, https://doi.org/10.1051/ANPHYS:0198400904060900
10. M. Yamanoi and J. H. Eberly, 'Relaxation terms for strong-field optical Bloch equations', Journal of the Optical Society of America B 1, 751-759 (1984). DOI: 10.1364/josab.1.000751, https://doi.org/10.1364/josab.1.000751
11. Z. Deng and S. Mukamel, 'Non-Markovian dephasing of two-level resonance fluorescence in a strong radiation field', Physical Review A 29, 1914-1921 (1984). DOI: 10.1103/physreva.29.1914, https://doi.org/10.1103/physreva.29.1914
12. G. Alber and J. Cooper, 'Two-photon collisional redistribution of radiation', Physical Review A 31, 3644-3671 (1985). DOI: 10.1103/physreva.31.3644, https://doi.org/10.1103/physreva.31.3644
13. K. Burnett, 'Collisional redistribution of radiation', Physics Reports 118, 339-401 (1985). DOI: 10.1016/0370-1573(85)90017-1, https://doi.org/10.1016/0370-1573(85)90017-1
14. S. Ya. Kilin and A. P. Nizovtsev, 'Generalised non-linear optical master equations taking into account the correlation time of relaxational perturbations', Journal of Physics B: Atomic and Molecular Physics 19, 3457-3475 (1986). DOI: 10.1088/0022-3700/19-21-013, https://doi.org/10.1088/0022-3700/19-21-013
15. C. Stéhlé and N. Feautrier, 'Absorption or emission during a collision: a test case H+2', Journal de Physique 47, 1015-1020 (1986). DOI: 10.1051/jphys:019860047060101500, https://doi.org/10.1051/jphys:019860047060101500
16. T. A. B. Kennedy and S. Swain, 'Dressed-atom approach to strong-field double-resonance fluorescence with laser phase fluctuations', Physical Review A 36, 1747-1758 (1987). DOI: 10.1103/physreva.36.1747, https://doi.org/10.1103/physreva.36.1747
17. M. Belsley and J. Cooper, 'Collisional redistribution by laser-induced fluorescence in the Ba-Ar system', Physical Review A 35, 1013-1021 (1987). DOI: 10.1103/physreva.35.1013, https://doi.org/10.1103/physreva.35.1013
18. J. Cooper, 'Destruction terms in non-Markovian collisional redistribution', Physical Review A 39, 409-412 (1989). DOI: 10.1103/physreva.39.409, https://doi.org/10.1103/physreva.39.409
19. S. M. R. Young, D. Segal, X. Chen and K. Burnett, 'Hg-Ar collisions in a strong laser field: competition between intense field and collisional effects', Journal of Physics B: Atomic, Molecular and Optical Physics 23, L791-L798 (1990). DOI: 10.1088/0953-4075/23-23-002, https://doi.org/10.1088/0953-4075/23-23-002
20. R. J. Bieniek, P. S. Julienne and F. Rebentrost, 'Collisional redistribution of polarized radiation for Sr-Ar(He) systems: a numerical comparison of the semiclassical decoupling/locking model to exact results', Journal of Physics B: Atomic, Molecular and Optical Physics 24, 5103-5119 (1991). DOI: 10.1088/0953-4075/24-24-010, https://doi.org/10.1088/0953-4075/24-24-010
21. S. Kryszewski and G. Nienhuis, 'The spectra of radiation redistribution in the presence of velocity-changing collisions', Journal of Physics B: Atomic, Molecular and Optical Physics 24, 3959-3979 (1991). DOI: 10.1088/0953-4075/24-18-011, https://doi.org/10.1088/0953-4075/24-18-011
22. M. Shah, S. M. R. Young, D. M. Segal and K. Burnett, 'Resonance atomic collisions in intense laser fields', Journal of Physics B: Atomic, Molecular and Optical Physics 25, 5383-5398 (1992). DOI: 10.1088/0953-4075/25-24-017, https://doi.org/10.1088/0953-4075/25-24-017
23. S. Swain, 'Exact and Approximate Rate Equations in Atom-Field Interactions', Advances in Atomic, Molecular, and Optical Physics 34, 345-377 (1994). DOI: 10.1016/S1049-250X(08)60023-3, https://doi.org/10.1016/S1049-250X(08)60023-3
24. P. R. Berman, 'Theory of fluorescence and probe absorption in the presence of a driving field using semiclassical dressed states', Physical Review A 53, 2627-2632 (1996). DOI: 10.1103/physreva.53.2627, https://doi.org/10.1103/physreva.53.2627
25. J. Szudy and W. E. Baylis, 'Profiles of line wings and rainbow satellites associated with optical and radiative collisions', Physics Reports 266, 127-227 (1996). DOI: 10.1016/0370-1573(95)00054-2, https://doi.org/10.1016/0370-1573(95)00054-2
26. F. Schüller and F. Rebentrost, 'Collisional redistribution of light: a comparative study of the density operator and optical collisional approaches', Zeitschrift für Physik D Atoms, Molecules and Clusters 37, 197-210 (1996). DOI: 10.1007/S004600050029, https://doi.org/10.1007/S004600050029
27. S. Y. Ni, W. Goetz, H. A. J. Meijer and N. Andersen, 'Far-wing collisional redistribution of light in the barium-rare gas systems', Zeitschrift für Physik D Atoms, Molecules and Clusters 38, 303-308 (1996). DOI: 10.1007/S004600050095, https://doi.org/10.1007/S004600050095
28. V. Bommier, 'The Density Matrix Theory for Polarized Radiation Redistribution', in Solar Polarization, Astrophysics and Space Science Library, 43-60 (1999). DOI: 10.1007/978-94-015-9329-8_4, https://doi.org/10.1007/978-94-015-9329-8_4
29. U. Vogl and M. Weitz, 'Laser cooling by collisional redistribution of radiation', Nature 461, 70-73 (2009). DOI: 10.1038/nature08203, https://doi.org/10.1038/nature08203
30. A. I. Parkhomenko and A. M. Shalagin, 'Kinetic equations for a density matrix describing nonlinear effects in spectral line wings', Journal of Experimental and Theoretical Physics 113, 762-771 (2011). DOI: 10.1134/S106377611114007X, https://doi.org/10.1134/S106377611114007X
31. J. Rosato, 'Collisional redistribution of hydrogen line radiation in low- and moderate-density magnetized plasmas', Physical Review E 103, 053209 (2021). DOI: 10.1103/PhysRevE.103.053209, https://doi.org/10.1103/PhysRevE.103.053209