## 2 Methods
### Transient Optical Properties with Pump-Probe Simulations  
Now, we describe the pump-probe simulation to investigate transient optical properties. Details of the numerical pump-probe simulation are described elsewhere [31].  
To compute transient optical properties of materials, we consider electron dynamics under pump \(\vec{E}_{pump}(t)\) and probe \(\vec{E}_{probe}(t)\) electric fields. Solving the time-dependent Kohn-Sham Equation (1) under the presence of both the pump and probe fields, one can simulate the electron dynamics induced by the fields. Furthermore, one can calculate the induced electric current by using Equation (3); we shall call the current under both the pump and probe pulses _pump-probe current_, \(\vec{J}_{pump-probe}(t)\). Additionally, one can simulate electron dynamics under only the pump pulse, and compute the current: we shall call the current induced solely by the pump pulse _pump current_, \(\vec{J}_{pump}(t)\). To extract the current induced by the probe pulse under the presence of the pump, we define the _probe current_, \(\vec{J}_{probe}(t)\), as  
\[\vec{J}_{probe}(t)\equiv\vec{J}_{pump-probe}(t)-\vec{J}_{pump}(t). \tag{6}\]  
Finally, referring to the above linear response equation, the transient conductivity can be evaluated as  
\[\sigma^{T}(\omega,T_{probe})=\frac{\int dtI_{probe}(t)e^{i\omega t-\gamma t}}{ \int dtE_{probe}(t)e^{i\omega t-\gamma t}}, \tag{7}\]  
Figure 1: (**a**) atomic configuration of monolayer \(h\)-BN; (**b**) current density in time domain after the impulsive distortion at \(t=0\); (**c**) real-part of surface conductivity, \(\Re\left[\sigma(\omega)\right]\), of \(h\)-BN, calculated by TDDFT with ALDA.  
where \(T_{probe}\) is the central time of the probe pulse. In contrast to the linear response calculation in equilibrium, the transient conductivity under the pump pulse depends also on the central time of the probe pulse since the time translation symmetry is broken by the pump. Thus, one may investigate dynamics of transient optical properties in time domain by changing the time delay between pump and probe pulses.  
To practically carry out the pump-probe simulations, we employ a pump vector potential of the following form:  
\[\vec{A}_{pump}(t) = -\frac{c\vec{E}_{pump}}{\omega_{pump}}\cos^{2}\left(\pi\frac{t} {T_{pump}}\right)\sin\left(\omega_{pump}t\right) \tag{8}\]  
in the domain \((-T_{pump}/2<t<T_{pump}/2)\) and zero outside of it. Here, \(\vec{E}_{pump}\) is the peak electric field, \(\omega_{pump}\) is the mean frequency, and \(T_{pump}\) is the full duration of the pump pulse.  
To perform the pump-probe simulation, we also employ a similar form for the probe vector potential:  
\[\vec{A}_{probe}(t) = -\frac{c\vec{E}_{probe}}{\omega_{probe}}\cos^{4}\left(\pi\frac{t- T_{delay}}{T_{probe}}\right)\sin\left[\omega_{pump}\left(t-T_{delay}\right)\right] \tag{9}\]  
in the domain \((-T_{probe}/2<t-T_{delay}<T_{probe}/2)\) and zero outside of it. Here, \(\vec{E}_{probe}\) is the peak electric field, \(\omega_{probe}\) is the mean frequency, and \(T_{probe}\) is the full duration of the probe pulse. The time delay between pump and probe pulses is expressed by \(T_{delay}\).