Ultrafast dissipative spin-state dynamics triggered by x-ray pulse trains

Frontiers of attosecond science are constantly shifting, thus addressing more and more intricate effects with increasing resolution. Ultrashort pulses offer a practical way to prepare complex superpositions of quantum states, follow, and steer their dynamics. In this contribution, an ultrafast spin-flip process triggered by subfemtosecond (fs) excitation and strong spin-orbit coupling between $2p$ core-excited states of a transition metal complex is investigated using density matrix-based time-dependent restricted active space configuration interaction theory. The effect of the nuclear vibrations is incorporated making use of an electronic system plus vibrational bath partitioning. The differences between isolated sub-fs pulses and pulse trains as well as the influence of various pulse characteristics on the initiated dynamics are discussed. The effect under study can be potentially used for ultrafast clocking in sub-few-fs experiments.

Figure 1

(a) Scheme explaining the parameters of the light pulse envelope [Eq. (4)]; see also Table 1. $T$ is the period of the driving laser optical cycle. (b) Calculated x-ray absorption spectrum. The energies corresponding to carrier frequencies $\hslash \mathrm{\Omega }=708.4$ and 711.5 eV are marked with arrows. The frequency combs corresponding to HHG with the respective wavelengths of the driving laser are also shown. The structure of the ${\left[\mathrm{Fe}{\left({\mathrm{H}}_2\mathrm{O}\right)}_6\right]}^{2+}$ complex is presented as an inset.

Figure 2

The cuts of the potential energy surfaces along the most active tuning vibrational mode (ground state frequency 417 ${\mathrm{cm}}^{-1}$). (a) and (c) Core and valence PESs of quintet SF states (red lines) and triplet SF states (blue lines), respectively, in the SF basis. (b) and (d) Same PESs in SOC basis, the color of lines corresponds to the collective contribution ${\sum }_{n,M_S}{\left|C_{an}^{(S=2,M_S)}\right|}^2$ of all quintet ($S=2$) SF states to a particular $|{\mathrm{\Psi }}_a^{\mathrm{SOC}}\rangle$, see Eq. (5). Note that only a part of the states is shown in (a) and (b), corresponding mainly to the $L_3$ absorption band. Also note that due to the hybrid approach (DFT ground state geometry and RASSCF PES) the minimum of the ground state PES is not at zero.

Figure 3

Huang-Rhys factors $g_{0n,\xi }^2/2$ for the transitions from the ground to the quintet (Q) and triplet (T) SF states for all 51 vibrational modes $\xi$. Inset: Distribution of relaxation rate $\hslash k_{a\to b}$ values [Eq. (12)] for different pairs ($a,b$) of valence and core SOC states. (Note that the first column of the histogram is cut at the value of 10.)

Figure 4

Population of selected states corresponding to the largest relaxation rates (cf. Fig. 3) with (dashed lines) and without (solid lines) bath: (a) core SF states, (b) valence SF states, (c) core SOC states, and (d) valence SOC states. The envelope of the excitation pulse is shown as a gray-shaded area.

Figure 5

Spin dynamics initiated by pulse trains having different characteristics: The shape of the normalized pulse envelope $\tilde{E}\left(t\right)/E_0={\sum }_{i}exp[-{(t-t_i)}^2/\left(2{\sigma }^2\right)]$ is depicted with gray filled curves. The populations of electronic SF states are presented for the cases of total quintet Q(tot) and triplet T(tot) as well as of the respective valence Q(val) and T(val) and core Q(core) and T(core). The black curve corresponds to the population of the lowest quintet (ground) state; note the finite temperature of 300 K resulting in an initial ground state population of 0.92. The amplitude $E_0$, wavelength of the driving laser $\lambda$, carrier frequency (energy) $\hslash \mathrm{\Omega }$, and subpulse duration $\sigma$ are given in the respective panels. The dissipation due to vibrational bath is included. (a) Dynamics initiated by a single pulse ($E_0=2.5\mathrm{a}.\mathrm{u}.$, $\hslash \mathrm{\Omega }=708.4\mathrm{eV}$, $\sigma =132$ as), no vibrational bath, no Auger decay. (b) Example of dynamics accounting for Auger decay. (c)–(f) Auger decay is excluded for simplicity of the analysis.

Figure 6

Spin dynamics initiated by pulse trains having different characteristics as given in the panels; see also Fig. 5.

Figure 7

Relative yield of the total triplet population T(tot) with respect to the total quintet one Q(tot) versus the integrated intensity $I_i={\int }_{-\infty }^{(t_{i+1}-t_i)/2}{\left|\tilde{E}\left(t\right)\right|}^{2}dt$. Here the time $(t_{i+1}-t_i)/2$ corresponds to the end of the $i\mathrm{th}$ subpulse in a train before $i+1\mathrm{th}$ subpulse starts. (a) and (b) $\lambda =800$ and 2000 nm. The population dynamics with/without Auger decay are shown as solid/dashed lines, respectively. For $E_0=2.5\mathrm{a}.\mathrm{u}.$ in (a) and (b), the first subpulse in the sequence is split into parts to ensure the overlap of two groups of curves along the $I_i$ axis, see text. The respective data are shown with filled circles. (c) Results for different $\lambda$ are presented for $\sigma =T/14$. Two horizontal lines denote the starting T(tot)/Q(tot) and the point of T(tot) = Q(tot).