Nature of the superconducting fluctuations in photoexcited systems

The photoexcited state associated with superconducting fluctuations above the superconducting critical temperature $T_c$ is studied based on the time-dependent Ginzburg-Landau approach. The excited state is created by an electric-field pulse and is probed by a weak secondary external field, which is treated by the linear response theory mimicking pump-probe spectroscopy experiments. The behavior is basically controlled by two relaxation rates: one is ${\gamma }_1$ proportional to the temperature measured from the critical point $T-T_c$, and the other is ${\gamma }_2$ proportional to the excitation intensity of the pump pulse. The excited state approaches the equilibrium state exponentially in a long time $t\gg {\gamma }_1^{-1}$, while in the intermediate-time domain we find a power-law or logarithmic decay with different exponents for $t\ll {\gamma }_2^{-1}$ and ${\gamma }_2^{-1}\ll t\ll {\gamma }_1^{-1}$, even though the system is located away from the critical point. This is interpreted as the critical point in equilibrium being extended to a finite region in the excited situation. The parameter dependences on both the pump and probe currents are also systematically studied in all dimensions.

Figure 1

Sketches for the time dependences of (a) the vector potential and (b) the electric field in the pump-probe process considered in the single pump-shot model. (c), (d) The same plots for the double pump-shot model.

Figure 2

Time evolution of the (a)–(c) superconducting fluctuation and the (d)–(f) excited current density in the single pump-shot model with a logarithmic scale, which are measured as deviations from the equilibrium component. The quantities are plotted for (a), (d) three-, (b), (e) two-, and (c), (f) one-dimensional systems. The dashed lines shown in (a), (c)–(f) are power functions shown in Eqs. (25, 26, 27, 28). $\mu$ is the direction where the pump pulse is shot, $\mu =z$ in the (d) three-, $\mu =y$ in the (e) two-, and $\mu =x$ in the (f) one-dimensional system. The normalized constants are defined by ${\psi }_0^2=\frac{C}{{\gamma }_1{\mathrm{\Gamma }}^{2}V}{\left(2m{\gamma }_1\mathrm{\Gamma }\right)}^{d/2}$ and $J_0=\frac{eC}{m{\gamma }_1{\mathrm{\Gamma }}^{2}V}{\left(2m{\gamma }_1\mathrm{\Gamma }\right)}^{\frac{d+1}{2}}$.

Figure 3

A sketch of the phase diagram in the plane of the logarithmic time $lnt$ and the logarithmic excitation strength $ln{\gamma }_2$. The dashed lines show crossovers.

Figure 4

Time dependence of the GL free energy defined in Eq. (12) in the single pump-shot model $\left(\delta {\mathcal{F}}_{\mathrm{tot}}=\delta {\mathcal{F}}_{\mathrm{fluct}}+\delta {\mathcal{F}}_{\mathrm{kin}}\right)$ in the (a) three-, (b) two-, and (c) one-dimensional systems, and we have taken ${\gamma }_2=100$. $\mu$ and $\nu$ mean the components of the kinetic GL free energy, where $\mu$ is the direction where the pump pulse is shot and $\nu$ is the other direction, $\mu =z$, and $\nu =x,y$ in the (a) three-dimensional system and $\mu =y$ and $\nu =x$ in the (b) two-dimensional system. The normalized constant is defined by ${\mathcal{F}}_0=\frac{C}{2\mathrm{\Gamma }}{\left(2m{\gamma }_1\mathrm{\Gamma }\right)}^{d/2}$.

Figure 5

Time dependences of (a) the superconducting fluctuation, (b) the excited current density, and (c) the GL free energy with the logarithmic time scale in the double pump-shot model. We have taken $t_0=1,{\gamma }_1=1$, and (c) ${\gamma }_2=100$ to compare with the results of the single pump-shot model in the two-dimensional system. $\mu$ and $\nu$ used in (c) have the same meanings as Fig. 4. The normalized constants are defined by ${\psi }_0^2=\frac{2mC}{\mathrm{\Gamma }V},J_0=\frac{2eC}{\mathrm{\Gamma }V}\sqrt{\frac{2m\mathrm{\Gamma }}{t_0}}$, and ${\mathcal{F}}_0=\frac{mC}{t_0}$.

Figure 6

(a) Time dependence of the nonequilibrium component of the dc paraconductivity, which is normalized by its equilibrium component ${\sigma }_{\mathrm{eq}}=\frac{e^{2}C}{2\pi Va}$. (b) Anisotropy parameter $\delta {\sigma }_{xx}/\delta {\sigma }_{yy}$. Note that the vertical axis in (b) is measured from unity (isotropic limit). We have taken ${\gamma }_1=1$ and $t_1=1.1$ with the time unit $t_0(=1)$.

Figure 7

Frequency dependence of the ac paraconductivity ${\sigma }^{\left(3\right)}$ at several different times. In these figures, we have taken ${\gamma }_1=1,{\gamma }_2=100$, and $t_1=1.1(t_0=1$ is the unit of time). The normalization constant is defined by ${\sigma }_0=\frac{8e^{2}C{t}_0}{\mathrm{\Gamma }V}$.

Figure 8

Frequency dependences of $L_1$ and $L_2$ in the (a) three-, (b) two-, and (c) one-dimensional systems.

Figure 9

Time dependences of the (a), (b) superconducting fluctuation, the (c), (d) excited current and the (e), (f) GL free energy with the logarithmic time scale in the double pump-shot model. The panels (a), (c), (e) are for three-dimensional systems and (b), (d), (f) for one-dimensional system. We take the parameter as ${\gamma }_1=1$ and (e), (f) ${\gamma }_2=100$ to compare with the results of the single pump-shot model $\left(t_0=1\right)$. $\mu$ and $\nu$ used in (e) have the same meanings as Fig. 4. The normalized constants are defined by ${\psi }_0^2=\frac{C{t}_0}{{\mathrm{\Gamma }}^{2}V}{\left(\frac{2m\mathrm{\Gamma }}{t_0}\right)}^{d/2},J_0=\frac{eC{t}_0}{m{\mathrm{\Gamma }}^{2}V}{\left(\frac{2m\mathrm{\Gamma }}{t_0}\right)}^{\frac{d+1}{2}}$ and ${\mathcal{F}}_0=\frac{C}{2\mathrm{\Gamma }}{\left(\frac{2m\mathrm{\Gamma }}{t_0}\right)}^{d/2}$.

Figure 10

Time dependences of the nonequilibrium component of (a) the dc paraconductivity and (b) anisotropy parameter $\delta {\sigma }_{xx}/\delta {\sigma }_{zz}$ in the three-dimensional systems (double pump-shot model). The vertical axis in (a) is normalized by the equilibrium component of the paraconductivity ${\sigma }_{\mathrm{eq}}=\frac{e^{2}C}{3\pi V}\sqrt{\frac{m}{a}}$. Note that the vertical axis in (b) is measured from the unity (isotropic limit). The dc paraconductivity for one-dimensional systems is also shown in (c), where the values are also normalized by ${\sigma }_{\mathrm{eq}}=\frac{e^{2}C}{2V\sqrt{m{a}^3}}$. We have taken ${\gamma }_1=1$ and $t_1=1.1$ with the time unit $t_0(=1)$.

Figure 11

Frequency dependence of the ac paraconductivity ${\sigma }^{\left(3\right)}$ at several different times in the (a) three- and (b) one-dimensional systems (double pump-shot model). In these figures, we have taken ${\gamma }_1=1,{\gamma }_2=100$, and $t_1=1.1$ with the unit time $t_0\left(=1\right)$. The normalization constant is defined by ${\sigma }_0=\frac{4e^{2}C}{m{\mathrm{\Gamma }}^{2}V}{\left(\frac{2m\mathrm{\Gamma }}{t_0}\right)}^{d/2}$.