# Photocontrollable Mixed-Valent States in Platinum-Halide Ladder Compounds
## 4 Photoexcitation and Following Lattice Relaxation
### Real-Time Dynamics
In addition, we estimate the absorbed photon density  
\[n_{\mathrm{ph}}=\frac{1}{4N}\frac{\langle\mathcal{H}_{\mathrm{HF}}(t=2.9 \mathrm{ps})\rangle-\langle\mathcal{H}_{\mathrm{HF}}(-\infty)\rangle}{\hbar \omega_{0}}, \tag{27}\]  
which indicates the effective intensity of pumping light. It monotonically increases with increasing \(A_{0}\). Figure 3 shows time evolution of the photoexcited IP CDW and OP CDW. On the IP-CDW background at \(t_{MM}^{\perp}/t_{MX}^{\parallel}=0.2\) (Figure 3a), local deformations are generated at around \(n=8\), 18, and 40 after photoirradiation, which correspond to STEs [42; 43]. The defect at \(n=40\) dissociates into solitons [44; 45; 46; 47], while the others just disappear. The solitonic excitation generates a stable antiphase CDW domain on the leg-1 chain. The IP-type charge-ordering pattern is thus photoconverted into the OP-type one. On the OP-CDW background at \(t_{MM}^{\perp}/t_{MX}^{\parallel}=0.2\) (Figure 3b), on the other hand, a local deformation appears at around \(n=38\) on the leg-2 chain after photoirradiation but it disappears soon. There occurs no phase conversion. Figure 4a shows the case of the IP-CDW background at \(t_{MM}^{\perp}/t_{MX}^{\parallel}=0.4\). Although the absorbed photon density is almost the same as the case of \(t_{MM}^{\perp}/t_{MX}^{\parallel}=0.2\), many soliton pairs appear. An interchain arrangement of the antiphase CDW domains becomes \"out of phase\" as time passes, which leads to the global phase conversion. On the OP-CDW background at \(t_{MM}^{\perp}/t_{MX}^{\parallel}=0.4\) (Figure 4b), on the other hand, several antiphase CDW domains are photoinduced but they do not contribute to the OP-to-IP phase conversion at all. An interchain arrangement of these domains is \"in phase\" and therefore there is no phase transition. This relaxation process reads as the formation of an antiphase OP-CDW (\(\overline{\mathrm{OP}}\)) structure. We here reveal light intensity dependences of the relaxation dynamics. Figure 5a-d presents OP- and IP-domain ratios,  
\[R_{\text{OP}}(t) =\frac{1}{2N}\sum_{n=1}^{N}\left(1-\frac{O_{n}(t)}{|O_{n}(t)|} \right), \tag{28}\] \[R_{\text{IP}}(t) =\frac{1}{2N}\sum_{n=1}^{N}\left(1+\frac{O_{n}(t)}{|O_{n}(t)|} \right), \tag{29}\]  
and normalized CDW amplitude,\[R_{\text{CO}}(t)=\frac{\sum_{n=1}^{N}\sqrt{|\tilde{d}_{n:1}(t)\tilde{d}_{n:2}(t)|} }{\sum_{n=1}^{N}\sqrt{|\tilde{d}_{n:1}(-\infty)\tilde{d}_{n:2}(-\infty)|}}, \tag{30}\]  
as functions of \(n_{\text{ph}}\). In both CDWs, charge ordering is weakened with increasing \(n_{\text{ph}}\). This is not surprising given that photon absorption reads as the charge-transfer excitation. We suppose that the CDW order is melted at around \(n_{\text{ph}}=0.11\)-\(0.14\) (a photon per 7-9 sites), where \(R_{\text{CO}}\) becomes less than \(0.2\) (shaded regions in Figure 5a-d). We focus on the CDW region. The naivest photoconversion processes by the charge-transfer excitations are illustrated in Figure 5e. These processes are expressed by the relation \(R\) (a conversion rate)\(=4n_{\text{ph}}\) (an absorbed-photon number per unit). On the IP-CDW background (Figure 5a,b), a weak photoexcitation can never induce the phase conversion. Once \(n_{\text{ph}}\) exceeds a threshold value, however, the conversion rate \(R_{\text{OP}}\) increases beyond \(R\). This can be regarded as a cooperative structural change due to a domino effect [48; 49]. Thus, the phase conversion is nonlinearly accelerated with increasing \(n_{\text{ph}}\), which is particularly pronounced in the strongly-coupled chains (Figure 5b). On the OP-CDW background (Figure 5c,d), on the other hand, the OP-to-IP phase conversion is never observed over the whole range of \(n_{\text{ph}}\). A decrease in the CDW amplitude shown in Figure 5 results in a decrease in the Coulomb energy difference between IP CDW and OP CDW in Equation (5). In such a situation, the interchain coupling \(t_{\text{MM}}^{\perp}\), which favors the OP arrangement, may become a more decisive factor in the phase competition.