ackerhalt_1977 ============== J\. R\. Ackerhalt and B\. W\. Shore, "Rate equations versus Bloch equations in multiphoton ionization", Phys\. Rev\. A 16, 277 \(1977\) `Source record `__ · :doc:`Catalogue ` · :doc:`Review register ` DOI: `10\.1103/PhysRevA\.16\.277 `__ **Source kind:** theory **Recorded technique:** The Wilcox\-Lamb reduction of the Bloch equations to rate equations \(the off\-diagonal density\-matrix elements set to their steady state\), shown for a two\-level atom with ionisation and collisional dephasing \(Eqs\. 1\-12\); numerical comparison of the Bloch and rate\-equation solutions for a resonant three\-level ladder ionised from its top level, for co\- and counterpropagating beams and a Doppler profile \(Eqs\. 13\-14, Figs\. 1\-9\)\. Model parameters only \(Rabi frequencies, ionisation and dephasing rates in units of the Rabi frequency\); no atom is named\. **Materials in structured entries:** None recorded. **Dataset quantities:** None recorded. **Contents:** 0 datasets; 0 formulas. Curation notes -------------- Nothing extracted\: a model study; it prints no atomic data \(Rule 7\: no noble\-gas number\)\. Pages are the journal's \(277\-282; PDF page n is page 276 \+ n\)\. Identity checked against the first page\: Physical Review A 16, No\. 1, July 1977, received 11 March 1977\. The library copy is a scan; Eqs\. 1\-12 were read from the page image \(p\. 278\)\. THE TWO\-LEVEL MODEL \(p\. 278\)\. Bloch equations with the off\-diagonal coupling \(1/2\) i Omega, Omega 'the on\-resonance Rabi frequency' \(Eqs\. 1\-3\), with damping put in 'in the standard way' \(Eqs\. 4\-6\)\: rho\_22 loses \(1/tau \+ R\) rho\_22; rho\_12 decays at 1/\(2 tau\) \+ R/2 \+ 1/\(2 T\_I\), with 1/tau the spontaneous rate, R the ionisation rate and 1/T\_I 'the collisional phase destruction rate'\. So the coherence decay is Gamma\_coh = R/2 \+ 1/\(2 T\_I\) \(with 1/tau = 0, as the paper takes\)\: half the population loss plus HALF of 1/T\_I\. In OEP 8's Eq\. 7 notation Gamma^coh = Gamma^PI/2 \+ gamma^phi, so gamma^phi = 1/\(2 T\_I\)\: Ackerhalt and Shore's 'phase destruction rate' is twice OEP 8's pure dephasing rate\. PRINTED SIGN INCONSISTENCIES \(no correction needed, nothing is taken\)\: Eqs\. 1 and 2 both print \+\(1/2\) i Omega \(rho\_12 \- rho\_21\), so they do not conserve population; Eq\. 5 has the minus sign for rho\_22\. Eq\. 3 has \+i Delta rho\_12 and Eq\. 6 \-i Delta rho\_12\. THE WILCOX\-LAMB RATE \(Eqs\. 7\-10\)\. With d rho\_12/dt = 0, rho\_11' = R\_s \(rho\_22 \- rho\_11\) and rho\_22' = \-R\_s \(rho\_22 \- rho\_11\) \- R rho\_22, with R\_s = \(1/4\) Omega^2 \(R \+ 1/T\_I\) / \[\(R/2 \+ 1/\(2 T\_I\)\)^2 \+ Delta^2\] \(Eq\. 10\), i\.e\. R\_s = \(Omega^2/2\) Gamma\_coh / \(Gamma\_coh^2 \+ Delta^2\)\. The photon flux \(Eq\. 11\) is printed as \(c/8 pi\) Omega^2/\(omega d^2\) and the cross section \(Eq\. 12\) as sigma = \(2 pi omega d^2 / c\)\(R \+ 1/T\_I\) / \[\(R/2 \+ 1/\(2 T\_I\)\)^2 \+ Delta^2\]\: Gaussian units with hbar absent from both \(hbar = 1; with Omega = d E\_0/hbar the flux is c hbar Omega^2/\(8 pi omega d^2\) and sigma carries 1/hbar\)\. Eqs\. 11 and 12 are consistent with each other as printed\. In SI the same result is sigma = \(omega d^2/\(eps\_0 hbar c\)\) Gamma\_coh / \(Gamma\_coh^2 \+ Delta^2\) \(derived 2026\-10\-09\)\. COMPARISON WITH OEP 8 \(derived 2026\-10\-09, not data\)\. Eq\. 10 is the one\-photon form of OEP 8 section 4\.5's rate\. Put OEP 8's two\-photon Rabi frequency Omega\_R = sqrt\(2 sigma\_0/pi\) Phi for Omega, Delta = Omega\_f \- 2 omega, and Gamma\_coh for the half\-width\: R\_s = \(Omega\_R^2/2\) Gamma\_coh / \(Gamma\_coh^2 \+ Delta^2\) = sigma\_0 Phi^2 \(Gamma\_coh/pi\)/\(Gamma\_coh^2 \+ Delta^2\) = sigma\_0 g Phi^2 with g the unit\-area Lorentzian in the two\-photon angular frequency\. So OEP 8's Omega\_R, its Gamma^coh and its rate W = sigma\_0 g Phi^2 are mutually consistent with the Wilcox\-Lamb limit, with the same factor\-of\-two conventions \(off\-diagonal Omega/2; coherence decay = half the population loss \+ pure dephasing\)\. VALIDITY OF RATE EQUATIONS \(Secs\. I, III\-V\)\. The rule quoted in Sec\. I\: rate equations hold when the coherent stimulated rates are much slower than the incoherent ones \(spontaneous emission, collisional phase interruption, ionisation\); the Wilcox\-Lamb rate equations are exact for very large 1/T\_I \(p\. 279\)\. The computed cases \(three\-level ladder, Omega\_1 = Omega\_2 = 1, R = 0\.5, constant fields, times in units of 1/Omega\)\: on resonance without collisions \(Fig\. 1\) the rate solutions 'represent the time\-averaged Bloch solutions'; with 1/T\_I = 1 \(Fig\. 2\) they agree after the initial transients; detuned by Delta = Omega = 1 per step with copropagating beams and no collisions \(Fig\. 3\) 'the rate equations do not predict any ionization, while the Bloch\-equation solutions \.\.\. predict roughly 20% ionization'; with 1/T\_I = 1 added \(Fig\. 4\) agreement is 'very good' after the transients \(about 65 % ionisation\); counterpropagating \(two\-photon resonant at every velocity, intermediate detuned by 1, Figs\. 5\-6\) agreement is very good\. Sec\. V\: with a collisional width of the order of the Rabi frequency the two solutions become identical after the transients\. Against OEP 8 section 4\.5\: Ackerhalt and Shore find agreement already at a dephasing comparable with the Rabi frequency, but only after the transients and only for constant\-amplitude fields; they do not test an envelope or a time\-dependent detuning\. Their failure case \(detuning comparable with the Rabi frequency, small dephasing\) is the regime a light shift comparable with Omega\_R puts a femtosecond pulse in, so it supports OEP 8's requirement that Gamma^coh also exceed the rates of change of the envelope and the light shift\. 'Agreement' here means the rate solution follows the time average of the Bloch one \(Sec\. I\), not the instantaneous populations\. FURTHER REFERENCES CITED, not in the library\: L\. R\. Wilcox and W\. E\. Lamb, Jr\., Phys\. Rev\. 119, 1915 \(1960\) \(the method\); B\. L\. Beers and L\. Armstrong, Jr\., Phys\. Rev\. A 12, 2447 \(1975\) \(two\-photon ionisation, Schrodinger and rate solutions identical at very weak or very strong intensity\); J\. R\. Ackerhalt and J\. H\. Eberly, Phys\. Rev\. A 14, 1705 \(1976\) \(multilevel ionisation ladders\)\. No two\-photon data are cited\. **Recorded bibliographic identity status:** VALIDATED\_WITH\_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents. Datasets -------- No datasets are recorded. Formulas -------- No formulas are recorded. Corrections ----------- No dataset errata are recorded in this source record. Formula corrections, where recorded, appear with their formulas above. Curation notes may also discuss corrections or unresolved interpretation. Consult the :doc:`review register ` for the checks and decisions.