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    Journals >Chinese Physics B >Volume 29 >Issue 8 >Page > Article
    • Chinese Physics B
    • Vol. 29, Issue 8, (2020)
    Multiphoton quantum dynamics of many-electron atomic and molecular systems in intense laser fields
    Peng-Cheng Li1、2、† and Shih-I Chu3
    Author Affiliations
  • 1Research Center for Advanced Optics and Photoelectronics, Department of Physics, College of Science, Shantou University, Shantou 55063, China
  • 2Key Laboratory of Intelligent Manufacturing Technology of MOE, Shantou University, Shantou 515063, China
  • 3Department of Chemistry, University of Kansas, Lawrence, Kansas 66045, USA
    • show less
    DOI: 10.1088/1674-1056/ab9c0f Cite this Article
    Peng-Cheng Li, Shih-I Chu. Multiphoton quantum dynamics of many-electron atomic and molecular systems in intense laser fields[J]. Chinese Physics B, 2020, 29(8): Copy Citation Text show less

    Abstract

    We present the recent new developments of time-dependent Schr?dinger equation and time-dependent density-functional theory for accurate and efficient treatment of the electronic structure and time-dependent quantum dynamics of many-electron atomic and molecular systems in intense laser fields. We extend time-dependent generalized pseudospectral (TDGPS) numerical method developed for time-dependent wave equations in multielectron systems. The TDGPS method allows us to obtain highly accurate time-dependent wave functions with the use of only a modest number of spatial grid point for complex quantum dynamical calculations. The usefulness of these procedures is illustrated by a few case studies of atomic and molecular processes of current interests in intense laser fields, including multiphoton ionization, above-threshold ionization, high-order harmonic generation, attosecond pulse generation, and quantum dynamical processes related to multielectron effects. We conclude this paper with some open questions and perspectives of multiphoton quantum dynamics of many-electron atomic and molecular systems in intense laser fields.

    Keywords

    $$ \begin{eqnarray}{\rm{i}}\displaystyle \frac{\partial \varPsi (X,t)}{\partial t}=\left[{H}_{0}+\boldsymbol A(t)\cdot \boldsymbol P+\displaystyle \frac{1}{2}N{\boldsymbol A}^{2}(t)\right]\varPsi (X,t),\end{eqnarray}$$(1)

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    $$ \begin{eqnarray}\boldsymbol P=\displaystyle \sum _{i=1}^{N}{\boldsymbol p}_{i}\end{eqnarray}$$(2)

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    $$ \begin{eqnarray}{H}_{0}=-\displaystyle \frac{1}{2}\displaystyle \sum _{i=1}^{N}{\boldsymbol p}_{i}^{2}+V=-\displaystyle \frac{1}{2}\displaystyle \sum _{i=1}^{N}{\nabla }_{{r}_{i}}^{2}+V,\end{eqnarray}$$(3)

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    $$ \begin{eqnarray}{\rm{i}}\displaystyle \frac{\partial {\varPsi }^{{\rm{V}}}(X,t)}{\partial t}=[{H}_{0}+\boldsymbol A(t)\cdot \boldsymbol P]{\varPsi }^{{\rm{V}}}(X,t).\end{eqnarray}$$(4)

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    $$ \begin{eqnarray}{\rm{i}}\displaystyle \frac{\partial {\varPsi }^{{\rm{L}}}(X,t)}{\partial t}=[{H}_{0}+\boldsymbol E(t)\cdot \boldsymbol R]{\varPsi }^{{\rm{L}}}(X,t),\end{eqnarray}$$(5)

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    $$ \begin{eqnarray}\boldsymbol R=\displaystyle \sum _{i=1}^{N}{r}_{i}\end{eqnarray}$$(6)

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    $$ \begin{eqnarray}\begin{array}{lll} & & {\rm{i}}\displaystyle \frac{\partial {\varPsi }^{{\rm{L}}}(r,t)}{\partial t}=[{H}_{0}+\boldsymbol E(t)\cdot \boldsymbol r]{\varPsi }^{{\rm{L}}}(\boldsymbol r,t),\end{array}\end{eqnarray}$$(7)

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    $$ \begin{eqnarray}\begin{array}{lll} & & {\rm{i}}\displaystyle \frac{\partial {\varPsi }^{{\rm{V}}}(\boldsymbol r,t)}{\partial t}=[{H}_{0}+\boldsymbol A(t)\cdot \boldsymbol p]{\varPsi }^{{\rm{V}}}(\boldsymbol r,t),\end{array}\end{eqnarray}$$(8)

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    $$ \begin{eqnarray}{H}_{0}=-\displaystyle \frac{1}{2}{\nabla }^{2}-\displaystyle \frac{Z}{r}.\end{eqnarray}$$(9)

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    $$ \begin{eqnarray}{H}_{0}=-\displaystyle \frac{1}{2}{\nabla }^{2}+V(r),\end{eqnarray}$$(10)

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    $$ \begin{eqnarray}\begin{array}{lll}{\rm{i}}\displaystyle \frac{\partial }{\partial t}{\psi }_{i\sigma }(\boldsymbol r,t) & = & H(\boldsymbol r,t){\psi }_{i\sigma }(\boldsymbol r,t)\\ & = & \left[-\displaystyle \frac{1}{2}{\nabla }^{2}+{V}_{{\rm{SIC}},\sigma }^{{\rm{OEP}}}(\boldsymbol r,t)\right]{\psi }_{i\sigma }(\boldsymbol r,t),\\ & & \,\,\,\,i=1,2,\ldots,{N}_{\sigma },\end{array}\end{eqnarray}$$(11)

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    $$ \begin{eqnarray}\rho (\boldsymbol r,t)=\displaystyle \sum _{\sigma }\displaystyle \sum _{i=1}^{{N}_{\sigma }}{\rho }_{i\sigma }(\boldsymbol r,t)={\rho }_{\uparrow }(\boldsymbol r,t)+{\rho }_{\downarrow }(\boldsymbol r,t).\end{eqnarray}$$(12)

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    $$ \begin{eqnarray}{V}_{{\rm{SIC}},\sigma }^{{\rm{OEP}}}(\boldsymbol r,t)={\upsilon }_{{\rm{H}}}(\boldsymbol r,t)+{\upsilon }_{{\rm{ext}}}(\boldsymbol r,t)+{V}_{{\rm{xc}},\sigma }^{{\rm{SIC}}}(\boldsymbol r,t),\end{eqnarray}$$(13)

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    $$ \begin{eqnarray}{\upsilon }_{{\rm{H}}}(\boldsymbol r,t)=\displaystyle \int \displaystyle \frac{\rho ({\boldsymbol r}^{^{\prime} },t)}{\boldsymbol r-{\boldsymbol r}^{^{\prime} }}{\rm{d}}{\boldsymbol r}^{^{\prime} }\end{eqnarray}$$(14)

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    $$ \begin{eqnarray}{V}_{{\rm{xc}},\sigma }^{{\rm{SIC}}}(\boldsymbol r,t)=\displaystyle \sum _{i}\displaystyle \frac{{\rho }_{i\sigma }(\boldsymbol r,t)}{{\rho }_{\sigma }(\boldsymbol r,t)}\left\{{\upsilon }_{i\sigma }(\boldsymbol r,t)+\left[{\bar{V}}_{{\rm{xc}},i\sigma }^{{\rm{SIC}}}(t)-{\bar{\upsilon }}_{i\sigma }(t)\right]\right\},\end{eqnarray}$$(15)

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    $$ \begin{eqnarray}\begin{array}{lll} & & {\upsilon }_{i\sigma }(\boldsymbol r,t)=\displaystyle \frac{\delta {E}_{{\rm{xc}}}^{{\rm{SIC}}}[{\rho }_{\uparrow },{\rho }_{\downarrow }]}{\delta {\rho }_{i\sigma }(\boldsymbol r,t)},\\ & & \end{array}\end{eqnarray}$$(16)

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    $$ \begin{eqnarray}\begin{array}{lll} & & {\bar{V}}_{{\rm{xc}},i\sigma }^{{\rm{SIC}}}(t)=\left\langle {\psi }_{i\sigma }\left|{V}_{{\rm{xc}},i\sigma }^{{\rm{SIC}}}(\boldsymbol r,t)\right|{\psi }_{i\sigma }\right\rangle,\\ & & \end{array}\end{eqnarray}$$(17)

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    $$ \begin{eqnarray}\begin{array}{lll} & & {\bar{\upsilon }}_{i\sigma }(t)=\langle {\psi }_{i\sigma }|{\upsilon }_{i\sigma }(\boldsymbol r,t)|{\psi }_{i\sigma }\rangle .\end{array}\end{eqnarray}$$(18)

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    $$ \begin{eqnarray}\varPsi (\boldsymbol r,t)=\displaystyle \sum _{l,m}\displaystyle \frac{{R}_{l}^{m}(r,t)}{r}{Y}_{l}^{m}(\theta,\phi ),\end{eqnarray}$$(19)

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    $$ \begin{eqnarray}{\rm{i}}\displaystyle \frac{\partial }{\partial t}{R}_{l}^{m}(r,t)=[{H}_{0}+V(r,t)]{R}_{l}^{m}(r,t).\end{eqnarray}$$(20)

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    $$ \begin{eqnarray}r(x)=L\displaystyle \frac{1+x}{1-x+\alpha },\end{eqnarray}$$(21)

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    $$ \begin{eqnarray}\left[{x}_{j}:\left(1-{x}_{j}^{2}\right){{P}^{^{\prime} }}_{N}({x}_{j})=0\right],\,(j=0,1,2,\ldots,N),\end{eqnarray}$$(22)

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    $$ \begin{eqnarray}{w}_{j}=\displaystyle \frac{2}{N(N+1)}\displaystyle \frac{1}{{[{P}_{N}({x}_{j})]}^{2}}.\end{eqnarray}$$(23)

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    $$ \begin{eqnarray}\begin{array}{c} & & {\phi }_{l}^{m}({x}_{j},t)={R}_{l}^{m}(r({x}_{j}),t)\displaystyle \frac{\sqrt{{r}^{^{\prime} }({x}_{j})}}{\sqrt{{P}_{N}({x}_{j})}}\sqrt{\displaystyle \frac{2}{N(N+1)}},\\ & & \,\,(j=1,2,\ldots,N-1),\end{array}\end{eqnarray}$$(24)

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    $$ \begin{eqnarray}{r}^{^{\prime} }(x)=L\left[\displaystyle \frac{2+\alpha }{{(1-x+\alpha )}^{2}}\right].\end{eqnarray}$$(25)

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    $$ \begin{eqnarray}{\phi }_{l}^{m}({x}_{j},t+\Delta t)={{\rm{e}}}^{-{\rm{i}}{H}_{0}^{l}\Delta t/2}{{\rm{e}}}^{-{\rm{i}}V(r,t)\Delta t}{{\rm{e}}}^{-{\rm{i}}{H}_{0}^{l}\Delta t/2}{\phi }_{l}^{m}({x}_{j},t).\end{eqnarray}$$(26)

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    $$ \begin{eqnarray}{{\rm{e}}}^{-{\rm{i}}{H}_{0}^{l}\Delta t/2}{\phi }_{l}^{m}({x}_{j},t)=\displaystyle \sum _{i}{S}_{ij}(l){\phi }_{l}^{m}({x}_{i},t),\end{eqnarray}$$(27)

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    $$ \begin{eqnarray}\displaystyle \sum _{i}{[{H}_{0}^{l}]}_{ij}{\chi }_{k}^{l}({x}_{i})={\varepsilon }_{k}^{l}{\chi }_{k}^{l}({x}_{j}),\end{eqnarray}$$(28)

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    $$ \begin{eqnarray}{S}_{ij}(l)=\displaystyle \sum _{k}{\chi }_{k}^{l}({x}_{i}){\chi }_{k}^{l}({x}_{j}){{\rm{e}}}^{-{\rm{i}}{\varepsilon }_{k}^{l}\Delta t/2}.\end{eqnarray}$$(29)

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    $$ \begin{eqnarray}{P}_{i\sigma }=1-{N}_{i\sigma },\end{eqnarray}$$(30)

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    $$ \begin{eqnarray}{N}_{i\sigma }=\langle {\psi }_{i\sigma }(\boldsymbol r,t)|{\psi }_{i\sigma }(\boldsymbol r,t)\rangle \end{eqnarray}$$(31)

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    $$ \begin{eqnarray}{D}_{i\sigma }(p,{\theta }_{p})={|{\mathop{\psi }\limits^{\sim }}_{i\sigma }^{{\rm{V}}}(\boldsymbol p,{t}_{{\rm{f}}})|}^{2},\end{eqnarray}$$(32)

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    $$ \begin{eqnarray}D(p,{\theta }_{p})={\left|\displaystyle \sum _{\sigma }\displaystyle \sum _{i=1}^{{N}_{\sigma }}{\mathop{\psi }\limits^{\sim }}_{i\sigma }^{{\rm{V}}}(p,{t}_{{\rm{f}}})\right|}^{2}.\end{eqnarray}$$(33)

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    $$ \begin{eqnarray}\begin{array}{ll}{V}_{l}= & -\displaystyle \frac{1}{r}-\displaystyle \frac{\alpha }{2{r}^{4}}{W}_{6}\left(\displaystyle \frac{r}{{r}_{{\rm{c}}}}\right)-\left(\displaystyle \frac{N-S}{r}+{A}_{1}\right){{\rm{e}}}^{-{B}_{1}r}\\ & -\left(\displaystyle \frac{S}{r}+{A}_{2}\right){{\rm{e}}}^{-{B}_{2}r},\end{array}\end{eqnarray}$$(34)

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    $$ \begin{eqnarray}{W}_{n}(x)=1-\left[1+nx+\displaystyle \frac{{(nx)}^{2}}{2!}+\cdots +\displaystyle \frac{{(nx)}^{n}}{n!}\right]{{\rm{e}}}^{-nx},\end{eqnarray}$$(35)

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    $$ \begin{eqnarray}{\rm{i}}\displaystyle \frac{\partial \psi (\boldsymbol r,t)}{\partial t}=[{H}_{0}+V(\boldsymbol r,t)]\psi (\boldsymbol r,t),\end{eqnarray}$$(36)

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    $$ \begin{eqnarray}{H}_{0}=-\displaystyle \frac{1}{2}{\nabla }^{2}+\displaystyle \sum _{l}|{Y}_{l}^{0}\rangle {V}_{l}\langle {Y}_{l}^{0}|,\end{eqnarray}$$(37)

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    $$ \begin{eqnarray}\begin{array}{ll}(t) & =\displaystyle \frac{{\partial }^{2}}{\partial {t}^{2}}\left\langle \psi (r,t)\left|z\right|\psi (r,t)\right\rangle \\ & =-\left\langle \psi (r,t)\left|\left[\hat{H},[\hat{H},z]\right]\right|\psi (r,t)\right\rangle .\end{array}\end{eqnarray}$$(38)

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    $$ \begin{eqnarray}S(\omega )=\displaystyle \frac{2}{3\pi {c}^{2}}{\left|\mathop{D}\limits^{\sim }(\omega )\right|}^{2},\end{eqnarray}$$(39)

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    $$ \begin{eqnarray}d(t)=\displaystyle \int z\rho (\boldsymbol r,t){\rm{d}}\boldsymbol r=\displaystyle \sum _{i\sigma }{n}_{i\sigma }\langle {\psi }_{i\sigma }(\boldsymbol r,t)|z|{\psi }_{i\sigma }(\boldsymbol r,t)\rangle,\end{eqnarray}$$(40)

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    $$ \begin{eqnarray}\begin{array}{lll} & {\nabla }^{2}{E}_{{\rm{L}}}(\boldsymbol r,t)-\displaystyle \frac{1}{{c}^{2}}\displaystyle \frac{{\partial }^{2}{E}_{{\rm{L}}}(\boldsymbol r,t)}{\partial {t}^{2}}= & {\mu }_{0}\displaystyle \frac{\partial {J}_{{\rm{abs}}}(\boldsymbol r,t)}{\partial t}\\ & & +\displaystyle \frac{{\omega }_{0}^{2}}{{c}^{2}}(1-{\eta }_{{\rm{eff}}}^{2}){E}_{{\rm{L}}}(\boldsymbol r,t),\end{array}\end{eqnarray}$$(41)

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    $$ \begin{eqnarray}{\nabla }^{2}{E}_{{\rm{H}}}(\boldsymbol r,t)-\displaystyle \frac{1}{{c}^{2}}\displaystyle \frac{{\partial }^{2}{E}_{{\rm{H}}}(\boldsymbol r,t)}{\partial {t}^{2}}={\mu }_{0}\displaystyle \frac{{\partial }^{2}[{P}_{l}(\boldsymbol r,t)+{P}_{nl}(\boldsymbol r,t)]}{\partial {t}^{2}},\end{eqnarray}$$(42)

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    $$ \begin{eqnarray}{P}_{l}(r,t)=\chi {E}_{{\rm{H}}}(r,t),\end{eqnarray}$$(43)

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    $$ \begin{eqnarray}{P}_{nl}(r,t)=[{n}_{0}-{n}_{{\rm{e}}}(r,t)]d(r,t),\end{eqnarray}$$(44)

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    $$ \begin{eqnarray}{P}_{{\rm{M}}}(\omega )\propto \displaystyle {\int }_{0}^{\infty }{{\rm{d}}}^{2}{r}_{\perp }{|{E}_{{\rm{H}}}({r}_{\perp },\omega )|}^{2}.\end{eqnarray}$$(45)

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    $$ \begin{eqnarray}{k}_{q}={k}_{{\rm{d}}}=q\nabla \phi (r,z,t)-\nabla \Phi (r,z,t),\end{eqnarray}$$(46)

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    $$ \begin{eqnarray}P(\omega )={\left|\displaystyle \frac{1}{({t}_{{\rm{f}}}-{t}_{{\rm{i}}}){\omega }^{2}}\displaystyle {\int }_{{t}_{{\rm{i}}}}^{{t}_{{\rm{f}}}}d(t){{\rm{e}}}^{-{\rm{i}}\omega t}{\rm{d}}t\right|}^{2}.\end{eqnarray}$$(47)

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    $$ \begin{eqnarray}I(t)={\left|\displaystyle {\int }_{{\omega }_{{\rm{i}}}}^{{\omega }_{{\rm{f}}}}P(\omega ){{\rm{e}}}^{{\rm{i}}\omega t}{\rm{d}}\omega \right|}^{2}.\end{eqnarray}$$(48)

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    $$ \begin{eqnarray}E(t)={E}_{1}{f}_{1}(t-{t}_{{\rm{d}}})\cos [{\omega }_{1}(t-{t}_{{\rm{d}}})+\phi ]+{E}_{2}{f}_{2}(t)\cos ({\omega }_{2}t),\end{eqnarray}$$(49)

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    Peng-Cheng Li, Shih-I Chu. Multiphoton quantum dynamics of many-electron atomic and molecular systems in intense laser fields[J]. Chinese Physics B, 2020, 29(8):
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