update $f_{n}$ to be $f_{n-1}$. The system temperature, $T_0$, is chosen such that the stationary distribution is gradually changing. Here, $T_0$ is set to be the mean value of the objective functions of all the perturbed samples. We keep track of how many times each profile was chosen at each iteration and generated a histogram of the occurrence frequency of each profile. We stop the sampling process when the histogram becomes stationary. This occurrence frequency is also called the likelihood of the transmission coefficient profiles. The higher the value of a profile’s likelihood is, the better the fit with the experimentally measured TDTR signals at different modulation frequencies. Thus by combining the PSO method with the Gibbs sampling algorithm, we are able to determine the most likely transmission coefficients at the interface between Si and Al.

Results
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Phonon transmission coefficients
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We demonstrate our transmission coefficient measurements on an Al film on Si substrate with the native oxide removed by Hydrofluoric acid prior to Al deposition, yielding a clean interface. The TEM image in Fig. \[fig:TDTRSignal\_clean\_300K\] (a) shows the interface thickness is less than 0.5 nm. The amplitude and phase of signals from the lock-in amplifier at different modulation 