Theory of coherent-oscillations generation in terahertz pump-probe spectroscopy: From phonons to electronic collective modes

Time-resolved spectroscopies using intense THz pulses appear as a promising tool to address collective electronic excitations in condensed matter. In particular, recent experiments showed the possibility to selectively excite collective modes emerging across a phase transition, as is the case for superconducting and charge-density-wave (CDW) systems. One possible signature of these excitations is the emergence of coherent oscillations of the differential probe field in pump-probe protocols. While the analogy with the case of phonon modes suggests that the basic underlying mechanism should be a sum-frequency stimulated Raman process, a general theoretical scheme able to describe the experiments and to define the relevant optical quantity is still lacking. Here we provide this scheme by showing that coherent oscillations as a function of the pump-probe time delay can be linked to the convolution in the frequency domain between the squared pump field and a Raman-like nonlinear optical kernel. This approach is applied and discussed in a few paradigmatic examples: ordinary phonons in an insulator, and collective charge and Higgs fluctuations across a superconducting and a CDW transition. Our results not only account very well for the existing experimental data in a wide variety of systems, but they also offer a useful perspective to design future experiments in emerging materials.

Figure 1

Scheme of the experimental configuration for the pump-probe protocol used, e.g., in Refs. [5, 6, 7]. The pump and the probe fields are both orthogonally incident on the sample, but they are polarized in two perpendicular directions with respect to the sample surface. The transmitted probe field is then recorded for a fixed observation time $t_{\mathrm{g}}$ as a function of the delay $t_{\text{pp}}$ between the two pulses.

Figure 2

Schematic of the mechanism responsible for the oscillations in $\delta E_{\text{probe}}\left(t_{\text{pp}}\right)$. Left: broadband pulse. When the pump duration is much shorter than the mode lifetime, panel (a), the Fourier transform of the pump field squared $A^2\left(\omega \right)$ is approximately constant around the mode resonance at ${\omega }_{\text{res}}$, panel (b). As a consequence, the convolution (10) between the pump and the mode is peaked at ${\omega }_{\text{res}}$, which identifies the frequency of the oscillations of $\delta E_{\text{probe}}\left(t_{\text{pp}}\right)$ in time domain, panel (c). Right: broadband pulse. In this case a longer pump pulse, panel (d), gives rise to a sharp $A^2\left(\omega \right)$ in the frequency domain, which can be even narrower than the mode itself, panel (e). When this condition is satisfied, the convolution $\delta E_{\text{probe}}\left(\omega \right)$ is peaked at $2{\mathrm{\Omega }}_{\text{pump}}$, which sets the frequency of the $\delta E_{\text{probe}}\left(t_{\text{pp}}\right)$ oscillations. If the pump frequency matches the resonance condition $2{\mathrm{\Omega }}_{\text{pump}}\approx {\omega }_{\text{res}}$, the amplitude of the oscillations is strongly enhanced, panel (f). Here $t_{\mathrm{g}}=0$ has been set in all the simulations.

Figure 3

Fermionic bubbles connecting the gauge field (wavy lines) with the phononic field (dashed lines). Solid lines denote the fermionic Green's function for the valence/conduction bands, as explicitly labeled in the diamagnetic diagrams in the top row. The vertexes identify the insertion either of a velocity $v_{ab}$ or a diamagnetic ${\tau }_{ab}$ term, which can have both intraband and interband components, with $a,b=v,c$ denoting valence and conduction bands, respectively.

Figure 4

Left: THz pump-eV probe detection of coherent phonon oscillations. Here the phonon mode oscillates at ${\omega }_0/\left(2\pi \right)=39.2$ THz, with spectral broadening $\gamma /\left(2\pi \right)=0.4$ THz. Panel (a) shows the time dependence of the pump and probe field (the observation time is set to $t_{\mathrm{g}0}=0$). Since the duration of the probe field is much shorter than the period of the phonon, there is no appreciable difference between the time evolution of the phonon displacement $Q\left(t\right)$ induced by the pump, panel (b), and the time dependence of the differential probe field as a function of the pump-probe time delay $t_{\text{pp}}$, panel (c). Right: THz pump-THz probe detection of coherent phonon oscillations. Panel (d) shows the time dependence of the pump and probe pulses. Panel (e) shows the differential transmitted probe field for three different values of $t_{\mathrm{g}}$, as marked in panel (d). Even though $\delta E_{\text{probe}}\left(t_{\text{pp}}\right)$ always oscillates at the phonon frequency, the choice of $t_{\mathrm{g}}$ affects the amplitude and phase of the signal.

Figure 5

(a) Pump and probe fields used to simulate the experiments of Ref. [5]. (b) Power spectrum of the squared pump field $A_p^2\left(\omega \right)$ compared to the nonlinear optical kernel $\left|K\right(\omega \left)\right|$ given by Eq. (36) for $\mathrm{\Delta }=0.65$ THz. (c) Differential transmitted probe field as a function of the pump-probe delay $t_{\text{pp}}$ obtained by fixing the observation time at $t_{\text{gate}}=0.1$ ps. (d) $\delta E_{\text{probe}}(t_{\mathrm{g}};t_{\text{pp}})$ as a function of both the observation time $t_{\mathrm{g}}$ and the time delay $t_{\text{pp}}$.

Figure 6

Time profile of the transmitted probe field in the SC case obtained within the attractive Hubbard model on the squared lattice at increasing filling $n=0.1,0.4,0.9$, and different values of the electronic broadening ($\gamma =0.01\mathrm{\Delta }$, left column; $\gamma =0.1\mathrm{\Delta }$, right column).

Figure 7

(a) Pump fields (red and green lines) used to simulate the experiments of Ref. [6] for two different central frequencies ${\mathrm{\Omega }}_i$ and ${\tau }_i$. The probe field (blue line) is the same of Fig. 5. (b) Power spectrum of the squared pump fields $A_i^2\left(\omega \right)$, compared to the nonlinear optical kernel $\left|K\right(\omega \left)\right|$ given by Eq. (36) for $\mathrm{\Delta }=0.65$ THz. (c) Differential transmitted probe field as a function of the pump-probe delay $t_{\text{pp}}$ obtained by fixing the observation time at $t_{\text{gate}}=0.1$ ps. For both choices of the pump, the oscillations are recovered at twice the pump frequency, with an enhanced amplitude when the resonance condition $2{\mathrm{\Omega }}_2\approx 2\mathrm{\Delta }$ is satisfied. (d) $\delta E_{\text{probe}}(t_{\mathrm{g}};t_{\text{pp}})$ as a function of both the observation time $t_{\mathrm{g}}$ and the time delay $t_{\text{pp}}$.

Figure 8

(a) Temperature dependence of the renormalized phonon frequency ${\mathrm{\Omega }}_0$ in units of the bare one ${\omega }_0$, as given by Eq. (48) at $T>T_{\text{CDW}}$ and Eq. (44) at $T<T_{\text{CDW}}$. (b) Raman kernel (41) of the amplitudon below $T_{\text{CDW}}$. Its value at the three selected temperatures shown by dashed lines is used to compute $\delta E_{\text{probe}}\left(t_{\text{pp}}\right)$ according to Eq. (10). (c) Differential field as a function of the pump-probe delay $t_{\text{pp}}$ for a broadband pulse with a central frequency of 0.9 THz and a duration of 0.8 ps.

Figure 9

Real (a) and imaginary (b) part of the kernel $K\left(\omega \right)$ as a function of $\omega /2\mathrm{\Delta }$. The continuous line corresponds to the finite broadening $\gamma =0.1\mathrm{\Delta }$ while the dotted line corresponds to $\gamma =0$. Here we used ${\omega }_D=10\mathrm{\Delta }$, which corresponds to the relatively weak dimensionless SC coupling ${\lambda }_{\text{SC}}\simeq 0.33$. (c),(d) Projection of the integration path of the integrals ${\int }_0^{\infty }ds\frac{e^{\pm is}}{\sqrt{s}}$ on the imaginary axis. The $+$ or $-$ signs appearing in the exponent of the integrand require us to choose the positive (c) or negative (d) imaginary axis, respectively. (e) Comparison between the function $K\left(t\right)$ (continuum blue line), as obtained by the numerical integration of Eq. (A2), and the long-time estimate of Eq. (A7) (dotted red line).