kaufman_2020
============

B\. Kaufman, T\. Rozgonyi, P\. Marquetand and T\. Weinacht, "Adiabatic elimination in strong\-field light\-matter coupling", Phys\. Rev\. A 102, 063117 \(2020\)

`Source record <https://github.com/LupoLab/optmat/blob/main/src/optmat/data/sources/kaufman_2020.toml>`__ · :doc:`Catalogue </the-data/sources>` · :doc:`Review register </register>`

DOI: `10\.1103/PhysRevA\.102\.063117 <https://doi.org/10.1103/PhysRevA.102.063117>`__

**Source kind:** theory

**Recorded technique:** Adiabatic elimination \(AE\) of an off\-resonant intermediate state from the time\-dependent Schrodinger equation of a three\-level model \(ground g, intermediate i, two\-photon\-resonant final e\), followed by the two\-photon rotating\-wave approximation \(TPRWA\)\: the effective two\-level equations with dynamic Stark shifts and a two\-photon Rabi frequency \(Eqs\. 1\-12, Appendix A\); the error of AE analysed by a second integration by parts \(Eqs\. 8\-9, 13\-14\) and by comparing numerical solutions of the full and the eliminated equations for Gaussian pulses of 3\-300 fs and one\-photon detunings of 50\-500 THz \(Figs\. 1\-8\)\. Model parameters \(Table II\) only; no atom or molecule is named\.

**Materials in structured entries:** None recorded.

**Dataset quantities:** None recorded.

**Contents:** 0 datasets; 0 formulas.

Curation notes
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Nothing extracted\: a model study; no noble\-gas number \(Rule 7\)\. Pages are the article's \(063117\-1 to \-7, PDF pages 1\-7\)\. Identity checked against the first page\: Physical Review A 102, 063117 \(2020\), published 23 December 2020\.

CONVENTIONS \(pp\. 1\-3\)\. The coupling is \<n\|H\_int\|m\> = \-\(1/2\) chi\_nm \(e^{\+i omega\_0 t} \+ e^{\-i omega\_0 t}\) with chi\_nm = mu\_nm E\(t\), E\(t\) the field envelope \(Eq\. 4\)\: the field is E\(t\) cos\(omega\_0 t\), hbar = 1, and E^\+ = \(E/2\) e^{\-i omega\_0 t} in OEP 8's convention\. After AE \(Eq\. 10\) and the TPRWA \(Appendix A, Table I\) the equations are \(Eqs\. 11, 12, A5, A6\)\: i c\_g' = omega\_g^S c\_g \+ Omega~ e^{\-i Delta\_2 t} c\_e and i c\_e' = omega\_e^S c\_e \+ Omega e^{\+i Delta\_2 t} c\_g, with omega\_g^S = \(chi\_gi^2/2\) omega\_gi/\(omega\_gi^2 \- omega\_0^2\), omega\_e^S = \(chi\_ei^2/2\) omega\_ei/\(omega\_ei^2 \- omega\_0^2\), Omega~ = \(1/4\) chi\_gi chi\_ei/\(omega\_ei \- omega\_0\), Omega = \(1/4\) chi\_ei chi\_gi/\(omega\_gi \+ omega\_0\), omega\_mn = omega\_m \- omega\_n and Delta\_2 = omega\_eg \- 2 omega\_0 \(p\. 5\)\. The matrix is Hermitian only at Delta\_2 = 0 \(Eq\. A6, p\. 6\)\. 'Omega\(t\)', called the two\-photon Rabi frequency, IS the off\-diagonal element\: half the Rabi frequency of OEP 8 and of kumar\_2026 \(whose off\-diagonal is Omega\_ge^P/2\)\. Appendix A's last sentence cites 'Eqs\. \(13\)' for the main\-text equations, which are Eqs\. \(11\)\.

COMPARISON WITH OEP 8 EQ\. 6 \(derived 2026\-10\-09, not data\)\. \(a\) Coupling\: OEP 8 has H\_fg = \-\(1/2\) M\_f \(E^\+\)^2 with, for one colour, M\_f = 2 sum\_n d\_fn d\_ng/\(E\_n \- E\_g \- hbar omega\) \(its Eq\. 2 at equal frequencies\), so \|H\_fg\| = sum\_n d d E\_0^2/\(4 \(E\_n \- E\_g \- hbar omega\)\)\. Kaufman et al\.'s hbar Omega = \-\(1/4\) mu mu E^2/\(E\_i \- E\_g \- hbar omega\_0\) \(omega\_gi = \-omega\_ig\)\: the same magnitude and sign, with the factor 1/4 = \(1/2\)\(2\)\(1/4\) accounted for\. \(b\) Stark shifts\: omega\_g^S = \-\(1/4\) E^2 \[2 mu\_gi^2 omega\_ig / \(hbar \(omega\_ig^2 \- omega\_0^2\)\)\] = \-alpha\_g \|E^\+\|^2 / hbar, with alpha\_g the full \(co\- and counter\-rotating\) dynamic polarisability of the one\-state model\: OEP 8's diagonal \-alpha\_j \|E^\+\|^2\. Same for e\. So Eqs\. 11/A5 are OEP 8's Eq\. 6 for one intermediate state, without ionisation or decay\. kumar\_2026's Eqs\. 7\-9 have exactly this form \(summed over the p states\), and Toth et al\.'s Eqs\. 6\-7 \(toth\_2023\) too\.

VALIDITY OF AE \(pp\. 1\-4, 6\-7\)\. The traditional condition is chi'/chi \<\< Delta \(the envelope's rate of change small against the one\-photon detunings; refs\. 1, 14\)\. The paper's sharper condition \(Eq\. 9\)\: \|\(c\_g' chi\_ig \+ chi\_ig' c\_g\)/\(chi\_ig c\_g\)\| \<\< \|omega\_g\-\|, and the same for the other three 'AE frequencies' omega\_{g,e \+\-} = omega\_{gi,ei} \+\- omega\_0 \(for Table II\: omega\_g\- = \-1000, omega\_g\+ = omega\_e\- = \-200, omega\_e\+\- = \+\-600 THz, p\. 4\)\. The neglected terms partly cancel \(between the two terms of the product rule, between the four AE frequencies, and in time, since chi' changes sign at the peak\), so AE 'works even when the envelope does not vary very slowly' \(p\. 4\)\: population error 2\.5 % for the Fig\. 1 case \(30 fs, 2/3 TW/cm^2, detuning 200 THz against a bandwidth of about 15 THz; the 'THz' are ordinary frequencies\); at constant pulse area the error grows as the pulse shortens from 300 fs \(no visible difference\) to 10 fs and 3 fs \('relatively close', Fig\. 2\); Fig\. 6 gives the error against detuning at 30 fs\. Table II\: E\_g = 0, E\_i = 600, E\_e = 800, nu\_0 = 400 THz, mu\_ig = 2\.11e\-29 C m, mu\_ie = 2\.09e\-29 C m, I\_0 = 0\.6667 TW/cm^2, tau\_I = 30 fs \(printed '\-tau\_I'; the cell has a stray dash\)\.

FINDING \(computed 2026\-10\-09 from Table II, not data\)\: integrating the full Eqs\. 5 and the eliminated Eqs\. 11 with the intensity FWHM 30 fs and I = \(1/2\) eps\_0 c E^2 gives P\_e = 0\.843 \(full\) and 0\.869 \(AE\), matching Fig\. 1 as drawn \(about 0\.845 and 0\.868\), a relative error of 3\.1 % \(2\.5 % printed\)\. The error scales with the peak intensity, not with the bandwidth\: at 30 fs it is 0\.64 %, 2\.2 % and 3\.1 % at 0\.0667, 0\.2 and 0\.6667 TW/cm^2, and at 0\.6667 TW/cm^2 and 100 fs it is 15 %\. In the weak\-field case it equals 2 \(chi/\(2 D\)\)^2 = 0\.64 % \(chi = mu E\_0/hbar, D = 200 THz x 2 pi, the intermediate admixture whose peak, about 0\.03 at 2/3 TW/cm^2, Fig\. 1 shows as P\_i\), while OEP 8's spectral bound \(its Eq\. 5, 2 kappa sigma^2 with kappa = 1/D^2 for one intermediate\) is 0\.20 %\. So the paper's constant\-area series \(intensity rising as the pulse shortens\) mixes two errors, and the larger one in its examples is the field\-strength \(fourth\-order\) correction neglected by the second\-order elimination, not the slowly\-varying\-envelope one\. OEP 8 section 4\.3 states only spectral conditions for the elimination; this case shows that \|d E\_0\|/\(2 hbar D\) \<\< 1 is also required \(it is small for helium at OEP 8's intensities and should be evaluated per gas\)\.

FURTHER REFERENCES CITED, not in the library\: refs\. 4\-17 on AE and multiphoton Rabi frequencies and Stark shifts \(e\.g\. C\. Trallero\-Herrero et al\., Phys\. Rev\. A 71, 013423 \(2005\); M\. Fewell, Opt\. Commun\. 253, 125 \(2005\); B\. J\. Sussman, Am\. J\. Phys\. 79, 477 \(2011\)\)\. No two\-photon data are cited\.

**Recorded bibliographic identity status:** VALIDATED. This concerns identification of the source; it does not establish the scientific correctness of its contents.

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No datasets are recorded.

Formulas
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No formulas are recorded.

Corrections
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Formula corrections, where recorded, appear with their formulas above. Curation notes may also discuss corrections or unresolved interpretation. Consult the :doc:`review register </register>` for the checks and decisions.