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function, and g$_{ν}$ the coupling matrix. 

By performing the Bogoliubov-de Gennes transforma- tion [21], c$_{k}$$_{↑}$ = u$_{k}$α$_{k}$ − v$_{k}$β † k and c$_{−}$$_{k}$$_{↓}$ = u$_{k}$β$_{k}$ + v$_{k}$α † $_{k}$, we obtain E$_{e}$ = ∑ $_{k}$E$_{k}$ ( 〈 α † $_{k}$α$_{k}$ 〉 − 〈 β$_{k}$β † k 〉 ), where E$_{k}$ = √ ξ 2 k + Δ 2 $_{k}$. The time evolution of 〈 α † $_{k}$α$_{k}$ and 〈 β$_{k}$β † k 〉 is calculated using the Heisenberg equations-of-motion ap- proach. They are found to be 

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and ∂ 〈 β$_{k}$β † k 〉 /∂τ = − ∂ 〈 α † $_{k}$α$_{k}$ 〉 /∂τ , where f$_{k}$ = f ( E$_{k}$ ) = 1 / ( e β$_{e}$E$_{k}$ + 1), N$_{Ω}$$_{0}$ = N (Ω$_{0}$) = 1 / ( e β$_{ph}$ Ω 0 − 1), δ$_{1}$ = δ ( E$_{k}$$_{−}$$_{q}$ − E$_{k}$ + Ω$_{0}$), and δ$_{2}$ = δ ( E$_{k}$$_{−}$$_{q}$ − E$_{k}$ Ω$_{0}$). Here we have specifically set Ω$_{ν}$ = Ω$_{0}$. Differentiation of both sides of the expression for E$_{e}$ and substitution of Eq. (2) leads to the rate of energy exchange 

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Recent experiment [19] shows that there exists cold lat- tice, which is negligibly coupled to the electrons but dis- sipates the energy of hot phonons via anharmonic cool- ing. Considering this observation, we write a set of rate equations for the electron, hot phonon, and lattice tem- peratures as 

where P$_{e}$ is the power for pumping electrons and P$_{ph}$ the power for pumping hot phonons. The specific heat of electrons can be calculated from the Boltzmann entropy S$_{e}$ = − 2 k$_{B}$ ∑ k { [1 − f ( E$_{k}$ )]ln[1 − f ( E$_{k}$ )]+ f ( E$_{k}$ )ln f ( E$_{k}$ ) } by C$_{e}$ = T$_{e}$∂S$_{e}$/∂T$_{e}$ , while the specific heat of hot phonons for one -vibrational mode can be calculated from the simple relationship C$_{ph}$ = ℏ Ω ∂N (Ω) /∂T$_{ph}$ | Ω=Ω $_{0}$. The results are given by 

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respectively, with coth x ≡ ( e x + e − $^{x}$) / ( e x − e − $^{x}$). Equa- tions (3)-(8) constitute our three-temperature model. 

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The original version of this model was phenomeno- logically proposed [19] as an extension of the two- temperature model [22] for the normal state. The present model has the following advantages: (i) it incorporates the detailed band structure, (ii) it is valid for both nor- mal and superconducting states, and (iii) it includes the anistropic effect on the el-ph coupling. 

Now we apply our three-temperature model to sim- ulate the time evolution of the electron, hot phonon, and lattice temperatures for a d -wave superconductor. We use a five-parameter tight-binding model [23] to describe the energy dispersion: ξ$_{k}$ = − 2 t$_{1}$ (cos k$_{x}$ + cos k$_{y}$ ) − 4 t$_{2}$ cos k$_{x}$ cos k$_{y}$ − 2 t$_{3}$ (cos 2 k$_{x}$ + cos 2 k$_{y}$ ) − 4 t$_{4}$ (cos 2 k$_{x}$ cos k$_{y}$ +cos k$_{x}$ cos 2 k$_{y}$ ) − 4 t$_{5}$ cos 2 k$_{x}$ cos 2 k$_{y}$ − µ , where t$_{1}$ = 1, t$_{2}$ = − 0 . 2749, t$_{3}$ = 0 . 0872, t$_{4}$ = 0 . 0938, t$_{5}$ = − 0 . 0857, and µ = − 0 . 8772. Unless specified explic- itly, the energy is measured in units of t$_{1}$ and the time is measured in units of ℏ /t$_{1}$ hereafter. (For t$_{1}$ = 150 meV, it corresponds to 1740 Kelvin in temperature and ℏ /t$_{1}$ to 4 . 4 femtosecond in time.) The d -wave gap func- tion has the form Δ$_{k}$ = Δ$_{0}$( T$_{e}$ )(cos k$_{x}$ − cos k$_{y}$ ) / 2. The temperature-dependent part is given by [24] Δ$_{0}$( T$_{e}$ ) = Δ$_{00}$tanh { ( π/z ) √ ar ( T$_{c}$/T$_{e}$ − 1) } , where z = Δ$_{00}$ / ( k$_{B}$T$_{c}$ ) and tanh x = 1 / coth x . In our calculations, we set Δ$_{00}$ = 0 . 2, the critical temperature T$_{c}$ = 0 . 06, the specific heat jump at T$_{c}$ is r = Δ C$_{e}$/C$_{e}$ ∼ 1 . 43, and a = 2 / 3. In this work, we focus on the buckling phonon mode, for which [16, 25, 26] g$_{B}$$_{1}$$_{g}$ = g$_{0}$ { [cos $^{2}$( q$_{x}$/ 2)+cos $^{2}$( q$_{y}$/ 2)] / 2 } − 1 / $^{2}${ φ$_{x}$ ( k ) φ$_{x}$ ( k + q )cos( q$_{y}$/ 2) − φ$_{y}$ ( k ) φ$_{y}$ ( k + q )cos( q$_{x}$/ 2) } with φ$_{x}$ = ( i/N$_{k}$ )[ ξ$_{k}$t$_{x,}$$_{k}$ t$_{xy,}$$_{k}$t$_{y,}$$_{k}$ ], φ$_{y}$ = ( i/N$_{k}$ )[ ξ$_{k}$t$_{y,}$$_{k}$ − t$_{xy,}$$_{k}$t$_{x,}$$_{k}$ ], N$_{k}$ = [( ξ 2 k − t 2 xy, $_{k}$) 2 + ( ξ$_{k}$t$_{x,}$$_{k}$ − t$_{xy,}$$_{k}$t$_{y,}$$_{k}$ ) 2 + ( ξ$_{k}$t$_{y,}$$_{k}$ − t$_{xy,}$$_{k}$t$_{x,}$$_{k}$ ) $^{2}$]1 / $^{2}$, t$_{α,}$$_{k}$ = − 2 t$_{1}$ sin( k$_{α}$/ 2), and t$_{xy,}$$_{k}$ = − 4 t$_{2}$ sin( k$_{x}$/ 2)sin( k$_{y}$/ 2). To be consistent with experi- ment [16, 19], we set g$_{0}$ = 0 . 4 and the mode frequency Ω$_{0}$ = 0 . 3. The relaxation time τ$_{β}$ = 200. The pump power is a Gaussian pulse of P = P$_{0}$e − τ $^{2}$/ (2 σ $^{$^{2}$)}$, for which, the corresponding FWHM (full width at half maximum) is 2.35 σ . To pump electrons, we set P$_{0}$ = 0 . 15 and σ = 1. Considering that the energy scale of phonons is much smaller than that of electrons, we set P$_{0}$ = 0 . 006 and σ = 100 for pumping phonons. The ratio C$_{ph}$/C$_{l}$ in Eq. (6) is set to be 0.2. 

