bassani_1977
============

F\. Bassani, J\. J\. Forney and A\. Quattropani, "Choice of gauge in two\-photon transitions\: 1s\-2s transition in atomic hydrogen", Phys\. Rev\. Lett\. 39, 1070\-1073 \(1977\)

`Source record <https://github.com/LupoLab/optmat/blob/main/src/optmat/data/sources/bassani_1977.toml>`__ · :doc:`Catalogue </the-data/sources>` · :doc:`Review register </register>`

DOI: `10\.1103/PhysRevLett\.39\.1070 <https://doi.org/10.1103/PhysRevLett.39.1070>`__

**Source kind:** theory

**Recorded technique:** Second\-order perturbation theory of the two\-photon transition 1s \-\> 2s of non\-relativistic hydrogen \(infinite nuclear mass, dipole approximation\), with the complete set of exact intermediate p states\: the discrete np summed term by term to N = 13 and extrapolated to N \-\> infinity, the continuum integrated numerically, the radial matrix elements from Bethe and Salpeter's closed forms \(their Eqs\. 63\.4, 71\.4, 71\.14\); in the length gauge J0 \(E\.r, Eq\. 1a\) and the velocity gauge J \(A\.p, Eq\. 1b\), the momentum matrix elements turned into those of r by p = \-\(i m / hbar\)\[r, H0\]\. Both photons polarised along the same direction\. Dimensionless frequencies omega\_i = 2 pi R nu\_i, nu\(1s\) = \-1, nu\(2s\) = \-1/4, R = 3\.29 x 10^15 sec^\-1 the Rydberg frequency \(pp\. 1070\-1072\)\.

**Materials in structured entries:** H

**Dataset quantities:** two\_photon\_transition\_amplitude

**Contents:** 2 datasets; 0 formulas.

Curation notes
--------------

Phys\. Rev\. Lett\. 39\(17\), 1070\-1073 \(24 October 1977\) = PDF pages 1\-4 \(journal page = PDF page \+ 1069\); a scan with a text layer\. Identity checked on the rendered pages on 2026\-10\-09\: the title, F\. Bassani \(Roma\), J\. J\. Forney and A\. Quattropani \(EPF Lausanne\), received 25 May 1977, on PDF p\. 1 \(journal p\. 1070, below the end of the preceding Letter\); headers 'VOLUME 39, NUMBER 17 PHYSICAL REVIEW LETTERS 24 OCTOBER 1977' and page numbers 1071\-1073 on PDF pp\. 2\-4\. Atomic hydrogen is the optmat material H\. Every number below was read on the rendered pages and the table images\.

THE AMPLITUDE \(Eqs\. 2\-5, pp\. 1070\-1072\)\. The rate is W\(1s\-2s\) = \|E1 E2\|^2 e^4 a0^4 / \(4 \(2 pi hbar^2\)^2 R^3\) \(1/9\) \|D\|^2 delta\(Delta nu\) \(Eqs\. 2a, 2b\), with D\[J0\] = \(3/2\) sum \(1 \+ P12\) e1\.\<2s\|r/a0\|n\>\<n\|r/a0\|1s\>\.e2 / \(nu\(n\) \- nu\(1s\) \- nu2\) \(Eq\. 3a\) and D\[J\] the same with p/\(m omega1 a0\) and p/\(m omega2 a0\) \(Eq\. 3b\), P12 interchanging the two photons\. For parallel polarisations along z only m = 0 p states enter, \<2s\|z\|np0\>\<np0\|z\|1s\> = R\_20^n1 R\_10^n1 / 3, and Eq\. \(3a\) becomes Eq\. \(4a\), D^d\[J0\] = \(1/2\) sum\_n \[\(1 \- 1/n^2 \- nu1\)^\-1 \+ \(1/4 \- 1/n^2 \+ nu1\)^\-1\] R\_10^n1 R\_20^n1, with the continuum Eq\. \(5a\)\. 'for polarization angle theta between the two photons, expressions \(4\) and \(5\) have to be multiplied by cos theta' \(p\. 1072\)\. The amplitude is quattropani\_1982's D\_1^n \(its Eq\. 3a\) for n = 2 and drachman\_1990's D1 \(its Eq\. 5\)\.

M\_ZZ PER UNIT \(derived here from Eq\. 3a, as quattropani\_1982 does from its Eq\. 3a\)\. The zz element of OEP 8's kernel \(its Eq\. 2\) is M\_zz = sum\_n \(1 \+ P12\) \<2s\|z\|n\>\<n\|z\|1s\> / \(E\_n1s \- omega2\) in atomic units \(hartree denominators\)\. Eq\. \(3a\) with e1 = e2 = z has the same numerators \(in units of a0^2\) and the denominators in rydberg, which are twice their values in hartree\: so sum\_n \(1 \+ P12\) z z / \(E\_n1s \- omega2\) = 2 x \(2/3\) D, M\_zz = \(4/3\) D, with the same sign\. The same factor holds for D\[J\], which Eq\. \(2b\) enters exactly as D\[J0\] enters Eq\. \(2a\)\. At nu1 = 3/8, delta^TP = \|M\_zz\|^2 = \(16/9\) D^2 = 246\.72 au for D = \-11\.7805 \(sigma\_0 = 2\.246e\-35 cm4 in TC\-952's convention, the value in quattropani\_1982's notes\)\.

THE RATE PREFACTOR \(finding\)\. Eq\. \(2\) is exactly the golden\-rule rate W = \(2 pi / hbar\) \|M\_fi\|^2 delta\(E\) for the paper's own definitions\: fields E cos omega t \(Eq\. 1a, so each absorbed photon brings e E / 2\), omega\_i = 2 pi R nu\_i with R the ordinary Rydberg frequency \(3\.29e15 s^\-1\), and delta\(E\) = delta\(Delta nu\) / \(h R\); checked numerically \(SI, ratio 1\.000000\)\. quattropani\_1982 prints the same prefactor in its Eq\. \(1\) but defines the fields as E0 \(e^{i omega t} \+ c\.c\.\) \(amplitude 2 E0\) and omega = R nu \(R angular\); under those definitions the golden\-rule rate is 16 x 8 pi^3 = 128 pi^3 times its Eq\. \(1\), which is the factor that quattropani\_1982's notes report\. So the 1982 prefactor is this paper's, carried over without the change of definitions; Eq\. \(2\) here is right as printed\.

TAKEN \(tools/extract/bassani\_1977\.py, from the library's CSV cells, checked against the table images\)\: the totals of Table I, D\[J0\] \(p\. 1071, tbl\_76d993024c8f; gauge length\) and of Table II, D\[J\] \(p\. 1072, tbl\_bb20405731ed; gauge velocity\), at nine nu1 from 0\.3750 to 0\.7475 \(rydberg, as printed\), under two\_photon\_transition\_amplitude, use = primary \(the paper's own exact calculation\), m\_zz\_per\_unit = 4/3 \(above\)\. No uncertainty is stated; the values are exact to their printing \(below\)\. Not taken\: the discrete partial sums \(N = 2\-13, infinity\) and the continuum parts, gauge\-dependent and not observables \(they enter the script's checks\)\.

CHECKED by the script, every one of the 270 printed cells \(15 in each of nine columns of both tables\)\: each equals the value computed here rounded \(248 cells\) or truncated \(22 cells\) to its printed digits, and none is off\. The partial sums from Eqs\. \(4a\), \(4b\) with Bethe and Salpeter's closed\-form radial integrals \(R\_20^21 = \-sqrt\(27\) the only negative one, as the paper says\); the sums to infinity by the same series to n = 4000 with its n^\-3 tail; the totals against the exact amplitude of quattropani\_1982's script \(a Dalgarno\-Lewis solution in Decimal arithmetic\)\: \-11\.780483, \-14\.731873, \-41\.148411, \-49\.687778, \-62\.659474, \-84\.525170, \-128\.683525, \-262\.165419, \-1334\.326138; the continuum parts against the exact amplitude less the exact discrete sum \(e\.g\. 1\.601895 and \-7\.088065 at 0\.3750\)\. The truncated cells include the J total at 0\.6875 \(\-49\.687778 printed '\-49\.6877', J0 '\-49\.6878'\) and both totals at 0\.7250 \('\-128\.683'\)\. Each total equals its printed parts' sum within 2 units of the last digit\.

THE GAUGE COMPARISON \(the paper's point, from the tables\)\. The totals agree in the two gauges at all nine nu1 \(to the printed digits; 0\.6875 differs only by the truncation above\), the parts do not\: the 2p term \(N = 2\) is \-17\.8785 in J0 \(152 % of the total at 0\.3750, 100\.8 % at 0\.7475\) and exactly 0 in J \(its factor 1/4 \- 1/n^2 vanishes\); the continuum is \+1\.60 to \+2\.14 in J0 \(\-14 % to \-0\.16 % of the total\) and \-7\.09 to \-651\.8 in J, 60\.2 % of the total at 0\.3750 falling to 48\.8 % at 0\.7475\. FINDING\: the text's 'more than half the total contribution in the A\.p gauge' \(p\. 1072\) holds for nu1 \<= 0\.7250 \(50\.8 %\), not at 0\.7375 \(49\.8 %\) and 0\.7475 \(48\.8 %\)\. Truncated at N = 13 without the continuum, J0 is 14\.1 % from the exact value at 0\.3750 and J 61\.1 %\: hence the paper's conclusion that E\.r converges faster and is 'the one to be used in any approximate calculation'\.

EQUATION MISPRINTS \(no taken value depends on them; findings\)\. \(a\) Eq\. \(4b\) prints the factor \[nu1 \(1/4 \- nu1\)\]^\-1; Table II's sums need \[nu1 \(3/4 \- nu1\)\]^\-1 \(nu2 = 3/4 \- nu1\), as Eq\. \(5b\) and the text's factor \(E\_2s \- E\_n\)\(E\_n \- E\_1s\) / hbar^2 omega1 omega2 \(p\. 1073\) have\: with 1/4 no N = 3 sum reproduces \(at 0\.3750 the 3p term is \-2\.7053 with 3/4, as printed, and \+8\.1 with 1/4\)\. Eq\. \(5b\) prints '\(3/4 \- nu\)' for \(3/4 \- nu1\)\. \(b\) Eq\. \(7\) \(p\. 1073\) prints '1/g' and 'epsilon^2 sqrt\(epsilon1\)'; Eqs\. \(2\), \(6\) and the photon density give 1/9 and epsilon2 sqrt\(epsilon1\) \(as Eq\. 8 has\), so f\(nu1\) = nu1 nu2 \|D\|^2 / 9 \(times the delta function\)\. The text's 'Introducing \(2\) and \(7\) into \(6\)' means the unnumbered photon\-density relation for '\(7\)'\.

NOT TAKEN\: Fig\. 1 \(p\. 1073\), the dimensionless absorption coefficient f\(nu1\) for 0\.675\-0\.750 \(a raster drawing of a function of the tabulated D, which the tables print; rule 8\); Eqs\. \(6\)\-\(8\)'s absorption coefficient, which needs the line shape \(the delta function\) and is not a frequency\-integrated two\_photon\_excitation\_cross\_section as printed\. The 2s lifetime is only mentioned \(Ref\. 5\), not computed\.

COMPARISONS\. drachman\_1990 \(TC\-1034\) reproduces this paper's 1s\-2s amplitudes as its Table IV column 'Ref\. 1' \(dataset ref1\_bassani\_1977, use = reproduced\)\: all nine values equal this paper's D\[J0\] as printed strings \(the script checks it\), including '\-49\.6878' at 0\.6875 where D\[J\] prints '\-49\.6877'; so the column is the length\-gauge row \(that dataset's gauge parameter says the column does not say which\)\. drachman\_1990's parenthesised 'Ref\. 1' discrete sums at 0\.3750 \(\-13\.3824, \-4\.6924\) are this paper's D^d at N \-\> infinity in J0 and J, and its first term, \-17\.87845, is the N = 2 entry \-17\.8785\. drachman\_1990's N0 = 15 D1 and D2 agree with this paper's totals to their printing\. quattropani\_1982's Table I\(a\) nu1 grid 0\.6750\-0\.7475 is this paper's \(checked\)\. Ref\. 2, J\. J\. Forney, A\. Quattropani and F\. Bassani, Nuovo Cimento B 37, 78 \(1977\) \(gauge invariance to all orders\), and Ref\. 5, J\. Shapiro and G\. Breit, Phys\. Rev\. 113, 179 \(1959\) and S\. Klarsfeld, Phys\. Lett\. 30A, 382 \(1969\) \(the 2s two\-photon lifetime\), are cited, not in the library \(checked by folder name\); Ref\. 7, T\. W\. Hansch, S\. A\. Lee, R\. Wallenstein and C\. Wieman, Phys\. Rev\. Lett\. 34, 307 \(1975\), observed 1s\-2s two\-photon excitation without an absorption value\.

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

Datasets
--------

bassani\_1977/D\_2s\_length
~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "D_2s_length",
     "material": "H",
     "file": "H/bassani_1977__D_2s_length.csv",
     "table": "tbl_76d993024c8f",
     "quantity": "two_photon_transition_amplitude",
     "unit": "1",
     "abscissa": "wavelength",
     "abscissa_unit": "rydberg",
     "use": "primary",
     "parameters": {
       "final_level": "2s",
       "gauge": "length",
       "polarization": "linear_parallel",
       "definition": "D[J0] = (3/2) sum_n (1 + P12) e1.<2s|r/a0|n><n|r/a0|1s>.e2 / (nu(n) - nu(1s) - nu2) (Eq. 3a), for parallel polarisations D^d[J0] = (1/2) sum_n [(1 - 1/n^2 - nu1)^-1 + (1/4 - 1/n^2 + nu1)^-1] R_10^n1 R_20^n1 (Eq. 4a) plus the continuum (Eq. 5a); R the radial matrix elements of r/a0, frequencies in rydberg, nu2 = 3/4 - nu1",
       "m_zz_per_unit": 1.3333333333333333,
       "method": "second-order perturbation theory, exact non-relativistic hydrogen eigenstates (infinite nuclear mass), discrete np summed to N = 13 and extrapolated, continuum integrated numerically"
     }
   }

Table I \(p\. 1071\), row D\[J0\], the total of the discrete sum \(N \-\> infinity\) and the continuum; nine nu1, 0\.3750\-0\.7475\. x and row\: nu1 = omega1 / \(2 pi R\), one photon's energy in rydberg as printed, the other 3/4 \- nu1\. Exact to its printing \(rounded, or truncated at 0\.7250\) against the exact amplitude \(tools/extract/bassani\_1977\.py\)\. No uncertainty stated\.

bassani\_1977/D\_2s\_velocity
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. code-block:: text

   {
     "id": "D_2s_velocity",
     "material": "H",
     "file": "H/bassani_1977__D_2s_velocity.csv",
     "table": "tbl_bb20405731ed",
     "quantity": "two_photon_transition_amplitude",
     "unit": "1",
     "abscissa": "wavelength",
     "abscissa_unit": "rydberg",
     "use": "primary",
     "parameters": {
       "final_level": "2s",
       "gauge": "velocity",
       "polarization": "linear_parallel",
       "definition": "D[J] = (3/2) sum_n (1 + P12) e1.<2s|p/(m omega1 a0)|n><n|p/(m omega2 a0)|1s>.e2 / (nu(n) - nu(1s) - nu2) (Eq. 3b), for parallel polarisations D^d[J] = -(1/2) sum_n [(1 - 1/n^2 - nu1)^-1 + (1/4 - 1/n^2 + nu1)^-1] R_10^n1 R_20^n1 (1/4 - 1/n^2)(1 - 1/n^2) [nu1 (3/4 - nu1)]^-1 (Eq. 4b, whose printed (1/4 - nu1) is a misprint) plus the continuum (Eq. 5b); frequencies in rydberg, nu2 = 3/4 - nu1",
       "m_zz_per_unit": 1.3333333333333333,
       "method": "second-order perturbation theory, exact non-relativistic hydrogen eigenstates (infinite nuclear mass), momentum matrix elements from those of r by p = -(i m / hbar)[r, H0], discrete np summed to N = 13 and extrapolated, continuum integrated numerically"
     }
   }

Table II \(p\. 1072\), row D\[J\], the total of the discrete sum \(N \-\> infinity\) and the continuum; nine nu1, 0\.3750\-0\.7475\. x and row\: nu1 as printed \(rydberg\), the other photon 3/4 \- nu1\. Exact to its printing \(rounded, or truncated at 0\.6875 and 0\.7250\) against the exact amplitude \(tools/extract/bassani\_1977\.py\); equal to D\[J0\] by gauge invariance\. No uncertainty stated\.

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 </register>` for the checks and decisions.