Formally linear response theory of pump-probe experiments

By linearizing the density of both the pump- and probe-excited states and neglecting the overlap between femtosecond laser pulses, the Kubo response theory is extended to describe pump-probe experiments. The main advantages of this response scheme is that although second order responses are included, it formally remains a linear theory and therefore all obtained expressions can be implemented straightforwardly within any standard band structure method, e.g., based on a Green’s function approach. In particular, even the time-dependent zeroth order dynamic conductivity as obtained by means of the spin-polarized relativistic screened Korringa-Kohn-Rostoker method for fcc Ni(100) predicts a relatively slow demagnetization process over 100 fs after the impact of the probe pulse, which is in reasonably good agreement with available experimental data.

Figure 1

Time-dependent normalized intensity resulting from two time-delayed $(\tau =20\mathrm{fs})$ identical, linearly polarized Gaussian femtosecond $(t_{\mathrm{FWHM}}=60\mathrm{fs})$ laser pulses (parallel configuration) neglecting the time overlap of the pulses.

Figure 2

Linear response to a femtosecond laser probe pulse as given by the dimensionless dynamic conductivity ${\tilde{\sigma }}_{\mu \nu }({\omega }_p,\omega )$ for $\mu ,\nu =x,y$ in the case of fcc Ni(100), when the pulse has a carrier frequency ${\omega }_p=2\mathrm{eV}$ (marked by a thin vertical line), a duration $\Delta t=60\mathrm{fs}$ (taken as full width at half-maximum, i.e., FWHM) and an envelope of a double exponential (full line), a Gaussian (dotted line), a hyperbolic secant (dashed line), and a Lorentzian shape (dotted-dashed line), respectively.

Figure 3

Strictly linear, frequency- and delay time-dependent (in grey for $\tau =50\mathrm{fs}$ and in black for $\tau =100\mathrm{fs}$) dimensionless zeroth order dynamic conductivity ${\tilde{\sigma }}_{\mu \nu }^{\left(0\right)}({\omega }_{\mathrm{pr}},\omega ;\tau )$ in the case of fcc Ni(100), when the probe pulse has a carrier frequency ${\omega }_{\mathrm{pr}}=2\mathrm{eV}$ (marked by a thin vertical line), a duration $\Delta t=60\mathrm{fs}$ (assumed as FWHM) and an envelope of a double exponential (full line), a Gaussian (dotted line), a hyperbolic secant (dashed line), and a Lorentzian shape (dotted-dashed line), respectively.

Figure 4

Gauss-windowed (damping factor $\alpha =3$) strictly linear, time- and delay time-dependent optical conductivity ${\tilde{\sigma }}_{\mu \nu }^{\left(0\right)}({\omega }_{\mathrm{pr}},t;\tau )$ as compared with ${\tilde{\mathcal{E}}}_{\mathrm{pr}}(t-\tau ){\tilde{\sigma }}_{\mu \nu }\left({\omega }_{\mathrm{pr}}\right)$ shown in grey, where ${\tilde{\mathcal{E}}}_{\mathrm{pr}}(t-\tau )$ is the envelope of the probe pulse (double exponential, full line; Gaussian, dotted line; hyperbolic secant, dashed line; and Lorentzian, dotted-dashed line) with a duration of $\Delta t_{\mathrm{FWHM}}=60\mathrm{fs}$ (thin vertical lines mark $t-\tau =\Delta t_{\mathrm{FWHM}}∕2$) and a carrier frequency of ${\omega }_{\mathrm{pr}}=2\mathrm{eV}$. Here ${\tilde{\sigma }}_{xx}\left({\omega }_{\mathrm{pr}}\right)=4.2036+1.2551i\left({10}^{15}\mathrm{Hz}\right)$ and ${\tilde{\sigma }}_{xy}\left({\omega }_{\mathrm{pr}}\right)=0.031305+0.050733i\left({10}^{15}\mathrm{Hz}\right)$, respectively.