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"# Two-photon absorption\n",
"\n",
"\n",
"In experimental studies of two- and multi-photon absorption processes, \n",
"the multiphoton transition strength, a function of all the\n",
"frequencies of all photons absorbed, is analyzed. For two\n",
"laser sources of circular frequency $\\omega_1$ and $\\omega_2$ with\n",
"associated wave lengths $\\lambda_1$ and $\\lambda_2$,\n",
"the\n",
"two-photon transition strength $\\delta (\\omega_1,\\omega_2)$\n",
"for the transition between states $\\mid 0 \\rangle $ and $ \\mid n \\rangle $ \n",
"in isotropic samples \n",
"is given by:\n",
"\n",
"\\begin{eqnarray*}\n",
"\\delta^{\\rm TPA}_{0n}(\\omega_1,\\omega_2) & = & {\\sf F} {\\mathcal{S}}_{\\alpha\\alpha}^{0n}(\\omega_1,\\omega_2)\n",
"{\\mathcal{S}}_{\\beta\\beta}^{on, *}(\\omega_1,\\omega_2) + {\\sf G} {\\mathcal{S}}_{\\alpha\\beta}^{0n}(\\omega_1,\\omega_2)\n",
"{\\mathcal{S}}_{\\alpha\\beta}^{0n, *}(\\omega_1,\\omega_2)+ \\\\ & & + {\\sf H} {\\mathcal{S}}_{\\alpha\\beta}^{0n}(\\omega_1,\\omega_2)\n",
"{\\mathcal{S}}_{\\beta\\alpha}^{0n, *}(\\omega_1,\\omega_2),\n",
"\\end{eqnarray*}\n",
"\n",
"where ${\\sf F}$, ${\\sf G}$ and ${\\sf H}$ are numbers depending on the polarization\n",
"state of the two photons and on the geometrical set-up (mutual\n",
"direction of the laser beams)\n",
"and,\n",
"in the dipole approximation,\n",
"\n",
"$$\n",
"{\\mathcal{S}}_{\\alpha\\beta}^{0n}(\\omega_1,\\omega_2) = \\frac{1}{\\hbar} \\sum_{m} \\Big \\{\n",
"\\frac{(\\hat{\\mu}_{\\alpha})_{0m} (\\hat{\\mu}_{\\beta})_{mn} } {\\omega_{m0} - \\omega_1}\n",
"+ \\frac{(\\hat{\\mu}_{\\beta})_{0m} (\\hat{\\mu}_{\\alpha})_{mn} } {\\omega_{m0} - \\omega_2} \\Big \\} \n",
"$$\n",
"\n",
"is the second-rank, two-photon tensor.\n",
"In the equation for $\\delta^{\\rm TPA}_{0n}(\\omega_1,\\omega_2)$ the summation runs over the whole set of\n",
"excited states, the energy conservation\n",
"relation $\\omega_1+\\omega_2=\\omega_{n0}$ applies,\n",
"and off-resonance conditions are implied - that is,\n",
"the frequencies $\\omega_1$ and $\\omega_2$ are sufficiently far off the\n",
"values at which the denominators vanish.\n",
"The tensor is non-symmetric in the exchange of the two\n",
"frequencies except for $\\omega_1=\\omega_2$.\n",
"For the special case of a one-colour beam - that is, a monochromatic light source\n",
" the transition matrix is symmetric and (using $\\omega=\\omega_1=\\omega_2$)\n",
"\n",
"$$\n",
"\\delta^{\\rm TPA}_{0n}(\\omega) = {\\mathcal{S}}_{\\lambda\\lambda}^{0n}(\\omega)\n",
"{\\mathcal{S}}_{\\mu\\mu}^{0n, *}(\\omega) + 2 {\\mathcal{S}}_{\\lambda\\mu}^{0n}(\\omega)\n",
"{\\mathcal{S}}_{\\lambda\\mu}^{0n, *}(\\omega)\n",
"$$\n",
"\n",
"for linear polarization of the incident light and\n",
"\n",
"$$\n",
"\\delta^{\\rm TPA}_{0n}(\\omega) = - {\\mathcal{S}}_{\\lambda\\lambda}^{0n}(\\omega)\n",
"{\\mathcal{S}}_{\\mu\\mu}^{0n, *}(\\omega) + 3 {\\mathcal{S}}_{\\lambda\\mu}^{0n}(\\omega)\n",
"{\\mathcal{S}}_{\\lambda\\mu}^{0n, *}(\\omega)\n",
"$$\n",
"\n",
"for circular polarization, respectively.\n",
"\n",
"It can be shown that the two-photon absorption\n",
"transition rate\n",
"(cross section) can be obtained from the single residue of the\n",
"cubic response\n",
"function.\n",
"\n",
"Two-photon absorption transition amplitudes\n",
"${\\mathcal{S}}_{\\alpha\\beta}^{0n}(\\omega_1,\\omega_2)$,\n",
"can also be extracted \n",
"from the single residue of a quadratic response function.\n",
"\n",
"The following relations hold\n",
"for the matrix elements of the two-photon transition tensor\n",
"\n",
"\\begin{eqnarray*}\n",
" S_{\\alpha\\beta}^{0n}(\\omega) & = & S_{\\beta\\alpha}^{0n}(\\omega_{n0} - \\omega), \\\\\n",
" S_{\\alpha\\beta}^{0n}(\\omega) & = & S_{\\alpha\\beta}^{n0}(- \\omega)^\\ast.\n",
"\\end{eqnarray*}\n",
"\n",
"For methods which do not fulfill this last equation (such as the \n",
"coupled cluster approach), one can instead use \n",
"the two-photon transition strength\n",
"\n",
"$$\n",
"{F}^{0n}_{\\alpha\\beta,\\gamma\\delta}(\\omega) = \\frac{1}{2} \\{ S_{\\alpha\\beta}^{0n}(-\\omega) S_{\\gamma\\delta}^{n0}(\\omega)\n",
" +S_{\\gamma\\delta}^{0n, *}(-\\omega) S_{\\alpha\\beta}^{0n, *}(\\omega)\\},\n",
"$$\n",
"\n",
"where $S_{\\alpha\\beta}^{0n}(\\omega)$ and $S_{\\alpha\\beta}^{n0}(\\omega)$ are the\n",
"left and right transition moments.\n",
"\n",
"Three-photon\n",
"absorption has been described through the\n",
"single residue of the cubic response function.\n",
"Similarly to two-photon absorption, one can discuss the\n",
"third-order (left and right) transition moments and strengths\n",
"and define the three-photon transition\n",
"strengths in this manner for\n",
"nonvariational wave functions.\n",
"\n",
"In general, the computational requirements for the calculation of\n",
"molecular properties from the residues of the response functions\n",
"inherit the requirements from the response functions themselves; that\n",
"is, the selection of the basis sets has to be done considering the\n",
"operators appearing in the expression for the transition moments and\n",
"excited-state properties. However, as the residues are connected to\n",
"specific excited states, the nature of the probed excited state \n",
"also needs to considered\n"
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