Terahertz time-domain spectroscopy of transient metallic and superconducting states

Time-resolved terahertz time-domain spectroscopy (THz-TDS) is an ideal tool for probing photoinduced nonequilibrium metallic and superconducting states. Here, we focus on the interpretation of the two-dimensional response function $\mathrm{\Sigma }(\omega ;t)$ that it measures, examining whether it provides an accurate snapshot of the instantaneous optical conductivity $\sigma (\omega ;t)$. For the Drude model with a time-dependent carrier density, we show that $\mathrm{\Sigma }(\omega ;t)$ is not simply related to $\sigma (\omega ;t)$. The difference in the two response functions is most pronounced when the momentum relaxation rate of photocarriers is small, as would be the case in a system that becomes superconducting following pulsed photoexcitation. From the analysis of our model, we identify signatures of photoinduced superconductivity that could be seen by time-resolved THz-TDS.

Figure 1

(a) Schematic illustration of the pulse sequence and time delays in a time-resolved THz-TDS measurement performed on the model system described in the text. The red lines represent the photoexcited carrier density that is generated by a pump pulse at $t=0$ and then decays on a time scale of $1/\mathrm{\Gamma }=20$ ps. The black lines represent the electric field of the THz probe pulse, shown for increasing time delay, $t_p$ (marked by $\times$), in the three panels. The blue lines depict the corresponding nonequilibrium current, and the blue dots mark its value at $t_s=2$ ps. The dotted line shows the impulse response function $\sigma (t_s-t)$ for a Drude metal. The truncation of the impulse response at $t=0$ differentiates $\mathrm{\Sigma }(\tau ,t_s)$ from $\delta \sigma (\tau ,t_s)$. (b) Nonequilibrum current as a function of $t_s-t_p$ for fixed $t_s=2$ ps and $1/\mathrm{\Gamma }=20$ ps. With increasing momentum relaxation time, ${\tau }_s$, a pulse of nonequilibrium current centered on $t_p=0$ grows in amplitude.

Figure 2

(a) Real and (b) imaginary parts of $\mathrm{\Sigma }(\omega ;t_s)$ as a function of $\omega$ for several values of ${\tau }_s$, with $t_s$ fixed at 1 ps and $\mathrm{\Gamma }=0$. The Drude conductivity for the same values of ${\tau }_s$ is shown as dotted lines for comparison. Spectra with different ${\tau }_s$ are normalized to ${\sigma }_0=n\left(0\right){\tau }_s{e}^2/m$ to illustrate the variation in frequency dependence. When $t_s/{\tau }_s$ is large, $\mathrm{\Sigma }(\omega ;t_s)$ asymptotically approaches the instantaneous (Drude) conductivity. For ${\tau }_s\ge t_s$ the temporal cutoff at $t=0$ generates oscillations, with period $\mathrm{\Delta }\nu =1/t_s$ along the frequency axis of $\mathrm{\Sigma }(\omega ;t_s)$.

Figure 3

Simulation of time-resolved THz-TDS response function for photocarriers whose dynamics are characterized by a time-independent scattering rate. (a) Imaginary part of $\mathrm{\Sigma }(\omega ;t_s)$, with ${\tau }_s=1$ ps and $1/\mathrm{\Gamma }=0.5$ ps, in both three-dimensional and contour plot representations. Contour levels indicate 0.01 steps from 0.01 to 0.1, inclusive. (b) Spectra of ${\mathrm{\Sigma }}_2(\omega ;t_s)$ (solid lines) for several values of $t_s$, with $\delta {\sigma }_2(\omega ;t_s)$ (dotted lines) shown for comparison. The spectrum for each $t_s$ is denoted by color in the legend; both ${\mathrm{\Sigma }}_2$ and ${\sigma }_2$ decrease with increasing $t_s$, as the carriers decay. All spectra are normalized to ${\sigma }_0\equiv \delta n\left(0\right)e^2{\tau }_s/m$.

Figure 4

Simulation of time-resolved THz-TDS response function for a model of transient photoinduced superconductivity. (a) Imaginary part of $\mathrm{\Sigma }(\omega ;t_s)$, with ${\tau }_s=1$ ps and $1/\mathrm{\Gamma }=0.5$ ps, in both three-dimensional and contour plot representations. Contour levels indicate 0.005 steps from $-0.01$ to 0.04, inclusive. (b) Spectra of ${\mathrm{\Sigma }}_2(\omega ;t_s)$ (solid lines) for several values of $t_s$, with $\delta {\sigma }_2(\omega ;t_s)$ (dotted lines) shown for comparison. The spectrum for each $t_s$ is denoted by color in the legend; with increasing $t_s, {\sigma }_2$ decreases uniformly as the carriers decay, while ${\mathrm{\Sigma }}_2$ shows oscillations about ${\mathrm{\Sigma }}_2\approx 0$, with period $\mathrm{\Delta }\nu =1/t_s$ and an amplitude that decays with both $\omega$ and $t_s$. All spectra are normalized to $\mathrm{\Delta }{\sigma }_0\equiv \mathrm{\Delta }n{e}^2{\tau }_s/m$.