Magneto-optical Kerr effect in pump-probe setups

TL;DR

The paper develops a general, efficient theoretical framework (DPOA) to compute time-resolved magneto-optical Kerr responses in ultrafast pump–probe experiments. By expressing the post-pump optical conductivity through the time-evolved SPDM and incorporating dissipative effects, it enables accurate long-time dynamics with reduced computational cost. The approach is demonstrated on a minimal four-band tight-binding model with spin–orbit coupling and Zeeman splitting and on weakly spin-polarized germanium, showing how Kerr rotation reveals n-photon resonances and pump-induced modifications to absorption. The framework provides a versatile tool for interpreting time-resolved MOKE measurements in realistic, multi-band materials and can be extended to more complex magnetic or topological systems.

Abstract

We develop a general theoretical framework for computing the time-resolved magneto-optical Kerr effect in ultrafast pump-probe setups, formulated within the Dynamical Projective Operatorial Approach (DPOA) and its application to the generalized linear-response theory for pumped systems. Furthermore, we exploit this formalism to express the post-pump optical conductivity and consequently the Kerr rotation in terms of the time-evolved single-particle density matrix (SPDM), providing a transparent and computationally efficient description of photo-excited multi-band systems. This extension, in addition to its lower computational cost, has the advantage of allowing the inclusion of phenomenological damping. We illustrate the formalism using both (i) a two-band tight-binding model, which captures the essential physics of ultrafast spin-charge dynamics and the Kerr rotation, and (ii) weakly spin-polarized germanium, as a realistic playground with a complex band structure. The results demonstrate that, by exploiting DPOA and/or its SPDM extension, one can reliably reproduce both the short-time features under the pump-pulse envelope and the long-time dynamics after excitation, offering a versatile framework for analyzing time-resolved magneto-optical Kerr effect experiments in complex materials. Moreover, this analysis clearly shows that the Kerr rotation can be used to deduce experimentally the relevant n-photon resonances for a given specific material.

Figures (5)

• Figure 1: (a) Schematic representation of the system, the pump pulse, and the time-delayed probe pulse together with its reflected component, showing the Kerr rotation of the reflected probe pulse polarization. The probe pulse arrives with a delay $t_{\mathrm{pr}}$ with respect to the pump pulse. The angle between the probe pulse polarization and the lattice vector is $\varphi$, which is set to $\varphi=0$ in our calculations. (b) Band structure of the equilibrium Hamiltonian including Rashba SOC and Zeeman splitting. The thickness of the solid lines on top of the dashed lines indicates the post-pump excitations, hole or electron, in VB or CB, respectively, for the pump pulse photon energy of $\unit[2.86]{eV}$. The double arrows mark the resonant inter-band energy gaps. (c) Real and imaginary parts of the equilibrium optical conductivities $\bar{\sigma}_{xx}^{\text{eq}}\left(\omega\right)$ and $\bar{\sigma}_{xy}^{\text{eq}}\left(\omega\right)$. Vertical gray solid lines mark the local extrema of $\theta_{K}^{\text{eq}}\left(\omega\right)$, see panel (d). (d) Equilibrium Kerr rotation angle, $\theta_{K}^{\text{eq}}\left(\omega\right)$. Vertical gray lines correspond to the local extrema of $\theta_{K}^{\text{eq}}\left(\omega\right)$, and vertical dashed lines indicate the pump photon energies used in the time-dependent simulations.
• Figure 2: (a--d) Real and imaginary parts of $\delta\bar{\sigma}_{xx}(\omega,t_{\mathrm{pr}})$ and $\delta\bar{\sigma}_{xy}(\omega,t_{\mathrm{pr}})$ as functions of $\omega$ and $t_{\mathrm{pr}}$. The horizontal dashed lines mark the pump pulse frequency $\hbar\omega_{\text{pu}}=\unit[2.86]{eV}$, while the vertical dashed lines indicate the probe pulse delay equal to the pump pulse FWHM, $t_{\mathrm{pr}}=\tau_{\mathrm{pu}}=\unit[10]{fs}$.
• Figure 3: (a--d) The evolution of the Kerr rotation angle, $\delta\theta_{K}(\omega,t_{\mathrm{pr}})$, for four different pump pulse frequencies indicated in Fig. \ref{['fig:system_eq']}(d). Horizontal dashed lines correspond to the respective pump pulse frequencies. Horizontal gray lines correspond to the local extrema of $\theta_{K}^{\text{eq}}\left(\omega\right)$, see Fig. \ref{['fig:system_eq']} (d).
• Figure 4: (a) Benchmark of the exact results for $\delta\theta_{K}(\omega,t_{\mathrm{pr}})$ against those obtained from the SPDM approach without damping ($\Upsilon=0$ in Eq. \ref{['eq:EOM_rho']}) for several probe pulse frequency cuts. The probe photon energy values of each cut, $\hbar\omega$, are given in eV. An offset of $\unit[0.015]{rad}$ has been applied on increasing the probe photon energies. The pump photon energy is $\hbar\omega_{\text{pu}}=\unit[2.86]{eV}$. (b) $\delta\theta_{K}(\omega,t_{\mathrm{pr}})$ at long delays without damping; the inset shows a cut at the resonant frequency $\omega=\omega_{\text{pu}}$. (c) Same as (b) but including damping in SPDM with $\Upsilon_{nm}=\delta_{nm}\tfrac{\lambda}{4}+(1-\delta_{nm})\tfrac{\lambda}{2}$. The horizontal solid and dashed lines in panels (b) and (c) are the same as those in Fig. \ref{['fig:d_Kerr']} (a).
• Figure 5: (a) Equilibrium Kerr rotation $\theta_{K}^{\text{eq}}\left(\omega\right)$ of weakly spin-polarized germanium. (b) $\delta\theta_{K}(\omega,t_{\mathrm{pr}})$ at long delays without damping; the inset shows a cut at the two-photon resonant frequency $\omega=2\omega_{\text{pu}}$. (c) Same as (b) but including damping in SPDM with $\Upsilon_{nm}=\delta_{nm}\tfrac{\lambda}{4}+(1-\delta_{nm})\tfrac{\lambda}{2}$. The horizontal gray and black dashed lines in all panels mark the one and two-photon resonances, $\hbar\omega=\hbar\omega_{\text{pu}}=\unit[1.55]{eV}$ and $\hbar\omega=2\hbar\omega_{\text{pu}}=\unit[3.10]{eV}$, respectively.