Photoinduced Enhancement of Excitonic Order

We study the dynamics of excitonic insulators coupled to phonons using the time-dependent mean-field theory. Without phonon couplings, the linear response is given by the damped amplitude oscillations of the order parameter with a frequency equal to the minimum band gap. A phonon coupling to the interband transfer integral induces two types of long-lived collective oscillations of the amplitude, one originating from the phonon dynamics and the other from the phase mode, which becomes massive. We show that, even for small phonon coupling, a photoinduced enhancement of the exciton condensation and the gap can be realized. Using the Anderson pseudospin picture, we argue that the origin of the enhancement is a cooperative effect of the massive phase mode and the Hartree shift induced by the photoexcitation. We also discuss how the enhancement of the order and the collective modes can be observed with time-resolved photoemission spectroscopy.

Figure 1

(a) Susceptibility ${\chi }_{11}^R(t)$ for different couplings $\lambda$. (b) $\lambda$ dependence of the frequencies of the collective modes at $q=0$ extracted from the peak positions in $-\mathrm{Im}{\chi }_{11}^R(\omega )$. The dashed black line indicates ${\omega }_0$.

Figure 2

(a)–(c) Time evolution of the gap at $k=0$, $\mathrm{\Delta }(k=0,t)$, the excitonic order parameter $|\varphi |$, and the phonon displacement $X$ for various field strengths and $e$-ph couplings. (d) $\lambda$ dependence of $|\varphi |$ averaged over $t\in [0,400]$, for various field strengths. The vertical line indicates the critical value ${\lambda }_c$ at $E_0=0.12$. (e),(f) Time evolution of $\mathrm{\Delta }(k=0,t)$ and $|\varphi |$ for $\lambda =0.1$ evaluated by freezing the Hartree shift. We have used $t_p=6.0$. The horizontal black line in each panel shows the corresponding equilibrium values.

Figure 3

(a) Trajectory of the normalized pseudomagnetic field ($B^x/B$, $B^y/B$) and the normalized pseudospin ($-S^x/S$, $-S^y/S$) at $k=0$ with and without phonons. (b) The time evolution of $B^x/B$ and $-S^x/S$ at $k=0$ with and without phonons. (c) The magnified trajectory of ($B^x/B$, $B^y/B$) and ($-S^x/S$, $-S^y/S$) and (d) the time evolution of $B^x/B$ and $-S^x/S$ at $k=0$ for $\lambda =0.1$. The parameters are $t_p=6.0$ and $E_0=0.12$. (e) Schematic picture of the trajectory of $\mathbf{B}/B$ (solid lines) and $-\mathbf{S}/S$ (dashed lines) with and without phonons. The equilibrium positions of $\mathbf{B}/B$ and $-\mathbf{S}/S$ are indicated by a black diamond in each panel.

Figure 4

(a),(b) trARPES spectra [$A_k^R(\omega ;t_{\mathrm{pr}})$] before (a) and after (b) the pump. Red dashed lines indicate the equilibrium quasiparticle dispersion from the mean-field theory. (c) Time evolution of the difference between the equilibrium and nonequilibrium gap size at each momentum $k$ [$\delta {\mathrm{\Delta }}_{\mathrm{ARPES}}(k,t_{\mathrm{pr}})$]. The orange solid line is $\delta {\mathrm{\Delta }}_{\mathrm{ARPES}}(0,t_{\mathrm{pr}})$. The parameters are $\lambda =0.1$, $E_0=0.18$, $t_p=160.0$, and ${\sigma }_{\mathrm{pr}}=12.0$. The black dashed line indicates the center of the pump pulse.