Time-dependent second-order Born calculations for model atoms and molecules in strong laser fields

Using the finite-element discrete variable representation of the nonequilibrium Green’s function (NEGF) we extend previous work [K. Balzer et al., Phys. Rev. A 81, 022510 (2010)] to nonequilibrium situations and compute—from the two-time Schwinger-Keldysh-Kadanoff-Baym equations—the response of the helium atom and the heteronuclear molecule lithium hydride to laser fields in the uv and xuv regimes. In particular, by comparing the one-electron density and the dipole moment to time-dependent Hartree-Fock results on one hand and the full solution of the time-dependent Schrödinger equation on the other hand, we demonstrate that the time-dependent second Born approximation carries valuable information about electron-electron correlation effects. Also, we outline an efficient distributed memory concept which enables a parallel and well-scalable algorithm for computing the NEGF in the two-time domain.

Figure 1

Distributed-memory concept in the case of (a) two MPI processes and (b) three MPI processes. The different domains with dotted, dashed, and long-dashed border (thin lines) mark main memory for $g^{\gtrless }(t,t^{\prime })$ allocated by distinct processes (ranks). The extent of the memory kernel (the right-hand side of the SKKBE) for the indicated time steps $T\to T+\delta T$ (cf. the arrows) is given by the thick dashed line, while the required self-energies ${\Sigma }^{\gtrless }(t,t^{\prime })$ are bordered with a thick solid line. Note that the relevant history is completely known to the process that performs the respective time step. In addition, the propagation of the greater and lesser Green’s function (extending the upper and lower triangle) can be performed by a single process.

Figure 2

Speed-up ratio $\mathcal{S}(p)=T_1/T_p$ [29] for parallel propagation of $G(1,1^{\prime })$ with up to $512$ MPI processes for three different propagation times $t_f=n_t\delta t$, where $\delta t=0.025$. As physical system, we have considered the one-dimensional He model (see Sec. 3) with $n_b=23$ FE-DVR basis functions. While the shortest calculations (plus symbols, red) have been performed on the “xe” nodes of the HLRN batch system [30], all other calculations have been carried out by the “ice1” nodes. We note that the performance drop between $4$ and $8$ processes is caused by internal architecture differences. The thin dashed, dotted, and dashed-dotted lines indicate a degree of parallelization of $\Gamma =95$% to $99$% according to Amdahl’s law ${\mathcal{S}}_{\Gamma }(p)$.

Figure 3

Contour plot (logarithmic) showing the time-dependent one-electron density $\langle \mathrm{n̂}\rangle (x,t)$ for 1D He in Hartree-Fock (solid, green) and second Born approximation (dashed, blue). The dashed-dot-dotted (red) lines reveal TDSE solutions. (a) Time evolution of the He atom initially prepared in the Hartree-Fock ground state and no uv field being switched on. (b) Short-time response of the Hartree-Fock initial state to a permanent uv field ($E_0=0.1$, ${\omega }_0=0.54$). (c) Permanent uv field as in (b) but with the He atom initially prepared in the self-consistent ground state. The contour lines cover a density range from ${10}^{-4}$ to $0.5$ a.u. with four contours in each decimal power between ${10}^{-4}$ and $0.1$, and contours between 0.1 and 0.5 each 0.05 a.u.

Figure 4

Nonlogarithmic plot of the 1D He density $\langle \mathrm{n̂}\rangle (x,t)$ for different times $t$ corresponding to Fig. 3c. The inset shows the self-consistent (initial) ground-state density at $t=0$. Further, the contour lines indicate the absolute of the density difference $\langle \delta \mathrm{n̂}\rangle {}_{\gamma }$ of TDHF (solid) and TD2ndB (dashed), respectively, to TDSE at values of $0.005$, $0.01$, $0.02$, $0.04,$ and $0.06$.

Figure 5

Time-dependent dipole moment $\langle \mathrm{d̂}\rangle (t)$ for the 1D helium atom. (a) TDHF (solid) and TD2ndB approximation (dashed-dotted) versus the exact TDSE result (dashed-dot-dotted). Note that the time evolution in case of TD2ndB starts from the Hartree-Fock ground state. (b) Comparison of the short-time behavior of $\langle \mathrm{d̂}\rangle (t)$ for different initial states: TD2ndB${}^2$ (dashed-dotted) refers to the second Born approximation with Hartree-Fock initial state, whereas the initial state for TD2ndB${}^1$ (dashed) has been computed self-consistently.

Figure 6

Dipole strength $\langle \mathrm{d̂}\rangle (\omega )$ for 1D He. (a) Discrete Fourier transform with Hamming window from TDHF (solid) and TDSE (dash-dotted) time series $\langle \mathrm{d̂}\rangle (t)$ of length $t_f=500$. The first excited state appears at ${\omega }_1=0.533$ (TDSE) and ${\omega }_1^{(1)}=0.549$ (TDHF); the second excited singlet state has frequency ${\omega }_2=0.671$. ${\mathcal{I}}_1=0.755$ denotes the first ionization threshold. Note that peaks beyond ${\mathcal{I}}_1$ are artificial box states. (b) Autoregression power spectral density $\langle \mathrm{d̂}\rangle {}_S(t)$ of short TDHF, TD2ndB, and TDSE time series $\langle \mathrm{d̂}\rangle (t)$ of length $t_f=55$ computed from Eq. (9) with model order $S=1000$. In the TD2ndB approximation, the first excited state appears at frequency ${\omega }_1^{(2)}=0.537$.

Figure 7

Time-dependent one-electron density $\langle \mathrm{n̂}\rangle (x,t)$ (plotted nonlogarithmically) for the response of LiH to an xuv field with intensity $E_0=0.75$ and frequency ${\omega }_0=1.3$. Shown are (a) the (exact) TDSE result and self-consistent (b) TD2ndB and (c) TDHF calculations. Arrows in (c) indicate particularly strong deviations of TDHF compared to TDSE.

Figure 8

One-electron density $\langle \mathrm{n̂}\rangle (x,t)$ at different times $t$ for the response of LiH to an xuv field (with parameters as in Fig. 7). The TDHF (solid) and TD2ndB${}^1$ result (dashed) include self-consistent initial states (with different but fixed $d_{\mathrm{b}}$) while TD2ndB${}^2$ (dotted) starts from the Hartree-Fock ground state (only shown for $t⩽10$). Note that the self-consistent solution of the TDSE (dashed-dot-dotted) is not fully accurate due to lack of spatial resolution—compare with the exact ground state (dashed-dotted line) for $t=0$. Moreover, the gray curves show the time-dependent expectation value $\langle \mathrm{x̂}\rangle (t)$ of the electron position.