\section{Introduction}

Failure in sheared metallic glasses (below the glass transition temperature $T_g$) generically occurs via localization of intense, irrecoverable deformation beyond its elastic limit, commonly termed as shear bands \cite{maass2015shear}.
These system-spanning strain features have typically narrow linear (in two dimensions) or planar (in three dimensions) topology along which most of plastic deformation accumulates, while the bulk of the material remains essentially elastically deformed. The underlying microstructural, dynamical mechanism that leads to shear-band nucleation and propagation in glassy metals has been known to follow a generally accepted picture in that the bulk plastic response emerges from \emph{collective} dynamics characterized by Shear Transformation Zones (STZs) \cite{argon1979plastic,falk1998dynamics, falk2011deformation, karimi2018correlation}.
Nevertheless, in realistic systems, such as metal alloys, this picture has been thought to be valid in the so-called large-atomic disparity limit, characterized by the misfit coefficient $\delta_a$~\cite{senkov2001effect}, with $\delta_a>0.06$ being the \emph{good}-glass-forming limit where the atomic radii of compositional elements are considered to be very different \cite{li2019mechanical}.
In this limit, local atomic displacements emerge spatially disordered, and under shear, they resemble the displacements seen in a wider range of soft glassy solids (i.e. jammed granular media, colloidal glasses, glassy polymers, or foams) within the context of plastic yielding transitions~\cite{ozawa2018random,denisov2017universal,karimi2017inertia,karimi2019plastic}.
Nonetheless, there is a way to generate a glassy environment in the opposite limit of $\delta_a \rightarrow 0$ but maintaining a larger number of elements, as in \add{medium/}high entropy metallic glasses (see \cite{ding2013high} and references therein). In this multicomponent limit, the disorder in atomic displacements emerges as well, but the origin is in the ``color" of the atoms, \add{meaning the complexity of enthalpic interactions}, as opposed to their size.
In this limit, the behavior under shear has been a challenging concept that remains unexplored. In this paper, we focus on several multicomponent alloys that have been considered in the literature as medium- and high-entropy alloys~\cite{li2018microstructures,li2019mechanical,frydrych2021materials}, and we compare for their mechanical properties in their corresponding glassy state, which we achieve by fast cooling~\cite{lu2017cooling}. In an effort to understand which element/compositional and microstructural features signify the quality of the emergent glass under fast cooling rates, we investigate the correlation between shear-band microstructures with the average stress-strain behavior, and then associate them to the compositional structure. We find that plastic rearrangements strongly correlate to the local atomic structure and the abundance of short-range ordering of icosahedra within rearranging zones increases glassy materials' capacity to delocalize strain. The misfit parameter $\delta_a$ is correlated to the overall compositional trend, but also microstructural parameters, such as the emergent shear-band width and shear-band dynamics' correlation, \add{as well as a bulk order parameter tied to the sharpness of plastic yielding transition}.

It is currently clear that almost every basic aspect of bulk metallic glasses deformation and failure (e.g. plasticity, shear localization, preparation dependence, strain rate sensitivity, and thermal effects) have been thoroughly understood based on the STZ mechanism \cite{schuh2007mechanical}.
Moreover, the general picture concerning the role of STZs in deforming glassy alloys has been further validated universally across.
This is remarkable given the inherent disparity of scales, microscopic interactions, and relaxation mechanisms involved.
Due to structural and/or mechanical) heterogeneity, the coalescence of atomic-scale STZs typically occurs in a highly intermittent and scattered manner both in time and space.
On approach to failure, STZ clusters restructure themselves over larger scales leading to the formation of co-existing macroscopic bands with a significant contribution to ductility and plastic flow \cite{wang2018spatial}.
On the other hand, certain (aged) glasses that lack this heterogeneity element \cite{fan2014evolution} (or associated lengths don't exceed interatomic scales) tend to localize strain within a single dominant band before the shear banding instablility results in a catastrophic brittle-type failure \cite{ma2016tailoring}.
Several studies highlighted the role of microstructural inhomogeneities in controlling the observed brittle-to-ductile transition.
In \cite{wang2008micromechanisms,das2005work,lee2006crystallization}, heterogeneities were mainly attributed to shear-induced crystallization introducing local hardening effects that hinder a dominant band propagation and, in turn, favor nucleation of multiple spontaneous shearing bands.
Similar properties were ascribed to quasi-crystal-like phases with short range order (SRO) featuring solid-like properties \cite{ding2014full,ding2012correlating,ding2014soft,ma2016tailoring} and, therefore, a strong tendency to tune the extent of strain localization \cite{shi2005strain,shi2006atomic,shi2007stress}.
SROs are indeed mediated by the formation of ordered icosahedral clusters as the most (energetically) favored atomic configuration in a quenched metallic glass which are also greatly responsible for a number of glassy properties such as the dynamical slow-down in the super-cooled regime, aging, and dynamic heterogeneity \cite{hufnagel2016deformation}.
We note that SROs serve as ``infertile" sites for the nucleation of STZs in that the latter are typically loosely-packed soft disordered arrangements that weaken local strength and enhance shear banding instabilities.

\begin{figure}[b]
    \centering
    \begin{overpic}[width=0.45\textwidth]{Figs/thermo.pdf}
        \LabelFig{13}{61}{$(a)$}
        \LabelFig{13}{10}{$(b)$}
	\end{overpic}
    \caption{Evolution of the per-atom enthalpy with temperature for CoNiCrFeMn upon \textbf{a}) melting \textbf{b}) quenching at $\dot T=0.1$~K/ps.  The insets show snapshots of atoms and associated radial distribution functions $g(r)$ at \textbf{a}) $T<T_m$ \textbf{b}) $T<T_g$. Here $T_m\simeq 1900$~K denotes the melting temperature corresponding to the enthalpy jump in \textbf{a}). The intersection of the nonlinear fits within the solid and liquid phases \cite{alcock1993materials} yields an estimate for the glass transition temperature $T_g\simeq 1000$~K in \textbf{b}).}
    \label{fig:thermo}
\end{figure}

\begin{figure
    \centering
    \begin{overpic}[width=0.3\textwidth]{Figs/loadCurve.png}
        \LabelFig{17}{81}{$(a)$}
        \Labelxy{52}{-3}{0}{\large $\gamma$}
        \Labelxy{-6}{35}{90}{{\large $\sigma_{xy}$\normalsize (Gpa)}}

        \LabelFigg{51}{16}{\tiny Co$_5$Cr$_2$Fe$_{40}$Mn$_{27}$Ni$_{26}$}
        \Labelxy{50.5}{16}{0}{\tiny Co$_5$Cr$_2$Fe$_{40}$Mn$_{27}$Ni$_{26}$}
 \end{overpic}

    \vspace{28pt}
    \begin{overpic}[width=0.23\textwidth]{Figs/mapGood.png}
        \LabelFig{-3}{100}{$(b)$}
        %
        \put(25,110){\includegraphics[height=0.15cm,width=1.7cm]{Figs/colorBar_new.png}}
        \put(25,103) {\footnotesize Low~~~~~~~High}

       \begin{tikzpicture
            \coordinate (a) at (0,0);
             \node[white] at (a) {\tiny.};
             \coordSys{3.7}{0}{.55}{1}{-0.6}{1}{1}{-0.2}{1}
        \end{tikzpicture}
    \end{overpic}
    %
    \begin{overpic}[width=0.23\textwidth]{Figs/mapBad.png}
        \LabelFig{-3}{100}{$(c)$}
       \begin{tikzpicture}
             \coordinate (a) at (0,0);
             \node[white] at (a) {\tiny.};

    	\end{tikzpicture}
	\end{overpic}
    \caption{\textbf{a}) Macroscopic stress $\sigma_{xy}$ plotted against applied (shear) strain $\gamma$ corresponding to several metallic glasses. Local (interpolated) squared nonaffine displacements maps associated with the \textbf{b}) Co$_5$Cr$_2$Fe$_{40}$Mn$_{27}$Ni$_{26}$ and \textbf{c}) CoNiCrFe glasses at $\gamma=0.2$. Here $x$, $y$, and $z$ denote flow, gradient, and vorticity directions, respectively.}
    \label{fig:loadCurve}
\end{figure}

The above big picture, however, fails to provide specific  details on the precise nature of SRO-induced inhomogeneities and clear description as to how structural heterogeneities evolve at atomic scales and control the shear banding dynamics on macroscopic levels.
In this framework, our study aims to establish robust microstructural origins associated with varying degrees of shear localization in driven glassy alloys.
Our approach will be based on two (qualitative) themes relevant to the nature of the plastic yielding transition and \emph{quality} of metallic glasses: \emph{\romannum{1}}) a ``good" glass with a high ability to form localized deformation patterns \emph{\romannum{2}}) a ``bad" glass with a tendency to delocalize strain.
\add{This might differ from the conventional notion based on the glass forming ability \cite{inoue2000stabilization} in that good glass formers (typically with notable $\delta_a$) do not necessarily deform well, according to our classification scheme.}
Our focus will be on certain types of local order which are purely structural in nature and strongly correlate with \emph{\romannum{1}}) dynamics of individual atoms \emph{\romannum{2}}) bulk stress response in good and bad glasses.
The structural metrics used in this study include the density of local icosahedral clusters as well as the radial distribution function of individual atoms.
The dynamical property is based on nonaffine displacements which should entail the atoms propensity to undergo localized rearrangements.
We also measure the spontaneous rate of stress release as a relevant mechanical property that helps discern a good glass from a bad one.
Using a numerical framework, our work addresses the intimate structure-dynamics-property relationship by probing the local atoms environment, morphology of nucleated shear bands, and macroscopic stress relaxation and seek for potential correlations with the composition-based misfit parameter in sheared metallic glasses.
The outcomes of our correlation analysis should describe the nature of the plastic yielding transition based upon relevant elemental/microstructural indicators with important implications in terms of materials tailoring to achieve desired mechanical properties.

The organization of the paper is as follows.
In Sec. \ref{sec:SimulationsAndProtocols}, the numerical setup, glass preparation, driving protocol, and relevant simulation details are discussed.
Section~\ref{sec:DiffuseVersusLocalized} presents the mechanical response corresponding to several simulated metallic glasses that, depending on chemical compositions, may nucleate diffuse or localized strain patterns.
In Sec. \ref{sec:StruDynProp}, we probe the structure-dynamics-property relationship through several structural and dynamical metrics.
Section~\ref{sec:discussions} and \ref{sec:conclusions} present discussions and conclusions, respectively.

\section{Simulations and Protocols}\label{sec:SimulationsAndProtocols}
Molecular dynamics simulations were carried out in LAMMPS \cite{plimpton1995fast} by implementing atomistic systems of size $N=50,688$ within periodic cubic boxes with $80\le L\le 90$~\r{A} in a simple-shear loading geometry.
The model metallic glasses included in this study are FeNi, CoNiFe, CoNiCrFe, CoCrFeMn, CoNiCrFeMn, and Co$_5$Cr$_2$Fe$_{40}$Mn$_{27}$Ni$_{26}$.
The interatomic forces were derived from the modified embedded atom method (MEAM) potential with the input parameters associated with the principle elements obtained from \cite{choi2018understanding}.
The glassy samples were prepared based on the \remove{standard} melt-and-quench protocol illustrated in Fig.~\ref{fig:thermo} plotting the enthalpy of CoNiCrFeMn as a function of temperature.
In this context, a \remove{(nearly)} crystalline structure was initially heated up to a high temperature ($3000$~K) \add{above the melting temperature $T>T_m$}, as in Fig.~\ref{fig:thermo}(a), \remove{far below $T_g$} before it was cooled down to the room temperature ($300$~K) below $T_g$ at the quench rate of $\dot{T}= 0.1$~K/ps, as in Fig.~\ref{fig:thermo}(b).
We allowed for further relaxation upon quenching for order $10$ ps to ensure as-quenched glasses are fully equilibrated prior to shearing.
The ambient pressure $P_0$ was set to zero at all times.
We also set the discretization time to $\Delta t=0.001$ ps.
The NPT ensembles were implemented via a Nose-Hoover thermostat and barostat with relaxation time scales $\tau_d^\text{therm}=0.2$~ps and $\tau_d^\text{bar}=2.5$~ps.
Prior to shearing, the bulk stress tensor $\sigma_{\alpha\beta}\simeq-P_0~ \delta_{\alpha\beta}$ has a nearly hydrostatic form.
A strain-controlled condition was applied by deforming the periodic box on the $xy$ plane \add{implementing Lees-Edwards boundary conditions \cite{lees1972computer}} at a fixed volume and temperature (canonical NVT ensembles) and a constant shearing rate $\dot\gamma=10^{-4}$ ps$^{-1}$ up to a $20\%$ strain.

\section{Deformation Patterns: Diffuse Vs. Localized}\label{sec:DiffuseVersusLocalized}
We performed a series of shear tests on the quenched samples with the resulting load curves $\sigma_{xy}$ against applied shear strain $\gamma$ reported in Fig.~\ref{fig:loadCurve}(a).
In all the simulated glasses, the stress rises monotonically toward a peak stress followed by a sheer reduction in strength as the loading continues.
We note that the Co$_5$Cr$_2$Fe$_{40}$Mn$_{27}$Ni$_{26}$ metallic glass exhibits a very pronounced and abrupt stress drop that is typically accompanied by sharp strain features as illustrated in Fig.~\ref{fig:loadCurve}(b).
The blue and red colors in the deformation maps indicate regions with low and high squared nonaffine displacements (see Sec.~\ref{sec:StruDynProp}).
The FeNi alloy almost shows identical stress features with an exception that the post-yielding behavior occurs in a smoother manner.
These metals should be referred to as good glasses with a high susceptibility to localize deformation.
The other compositions, the so-called bad glasses, seem to indicate a shallower stress decay that is associated with rather diffuse deformation patterns across the medium, as in Fig.~\ref{fig:loadCurve}(c).

Intuitively, the blue matrix in the deformation maps serve as fertile spots for the nucleation of SROs whereas the red (hot) regions should be structurally rich in terms of STZ clusters.
Yet, there is an evolving fraction of icosahedral arrangements that tend to co-exist with STZs within the hot spots.
We argue that the observed co-existence indeed contributes to structural heterogeneities associated to the sheared glass and, effectively, controls its quality (i.e goodness versus badness) upon failure.

\begin{figure}[b]
    \centering
    \begin{overpic}[width=0.3\textwidth]{Figs/hminIllustration.png}
        \Labelxy{50}{-3}{0}{\large $\gamma$}
        \Labelxy{-6}{35}{90}{\large $\sigma_{xy}$\normalsize (Gpa)}
       \begin{tikzpicture}
            \coordinate (a) at (0,0);
            \node[white] at (a) {\tiny.};

                \drawSlope{3.0}{3.4}{0.4}{20}{black}{\hspace{-8pt}\large h_\text{min}}{\vspace{2pt}1}

			\coordinate (center2) at (3.1,3.8);
            \coordinate (b) at ($ (center2) - .3*(1,-2.3) $);
            \coordinate (c) at ($ (center2) + .3*(1,-2.3) $);
            \draw[black][line width=0.4mm] (c) -- (b);
            \coordinate (d) at ($ (center2) - .5*(0,1) $);
		    \end{tikzpicture}

	\end{overpic}
    \caption{Illustration of the softening modulus $h_\text{min}$. The (black) crosses indicate the stress data $\sigma_{xy}$ as a function of the applied (shear) strain $\gamma$. The dashdotted (red) curve denotes a nonlinear fit based on (cubic) smoothing splines.  The square symbol indicates the strain corresponding to the minimal slope past the peak as illustrated by the straight line.}
    \label{fig:hminIllustration}
\end{figure}

\section{Structure-dynamics-property relation}\label{sec:StruDynProp}
We now turn to relevant structural and dynamical metrics and search for possible connections with glass properties.
As a starting point, dynamics of individual atoms is quantified by $D^2_\text{min}$ as a measure of deviations from affine trajectories of atoms imposed by an external (homogeneous) shear \cite{falk1998dynamics}.
We fit a linear model based on the reference positions $\vec{\mathring{r}}$ at zero strain, i.e $\vec{\mathcal{R}}(\vec{\mathring{r}})={\bf{F}} \vec{\mathring{r}}+\vec{F}_0$, to the current atomic positions $\vec{r}$ associated with $N_b$ atoms within a given volume \footnote{The $D^2_\text{min}$ analysis involves partitioning of the entire box into sub-volumes using a cubic grid of size $r_c=3.0$~\r{A} which is close to twice the radius associated with the first peak of $g(r)$.} by minimizing the squared loss function $\mathcal{L}({\bf{F}},
\vec{F}_0)=\sum_{i=1}^{N_b}|\vec{r}_i-\vec{\mathcal{R}}(\vec{\mathring{r}}_i)|^2$ to obtain $\{\hat{\bf{F}},\hat{\vec{F}}_0\}=\text{argmin}~\mathcal{L}({\bf{F}},
\vec{F}_0)$.
Here the second-ranked deformation tensor $\bf{F}$ and the (rigid) translation vector $\vec{F}_0$ correspond to an affine transformation.
The squared nonaffine displacements for atom $i$ is defined as $D^2_{i}=|\vec{r}_i-\hat{\vec{r}}_i|^2$ with $\hat{\vec{r}}_i$ being the predicted position $\hat{\vec{\mathcal{R}}}(\vec{\mathring{r}})={\hat{\bf{F}}} \vec{\mathring{r}}+\hat{\vec{F}}_0$ due to affine deformations.

Next, we performed a Voronoi analysis using OVITO \cite{stukowski2009visualization} to locate atoms with an icosaheral symmetry---that is, Voronoi cells with exactly 12 polyhedral faces with 5 edges each.
To obtain the associated (number) density, $\rho_i^\text{ico}=1/V_i^\text{ico}$, we repeated the Voronoi analysis by including \emph{exclusively} atom $i$'s with icosahedral symmetries within the periodic box (and excluding the other atoms).
This gives another set of Voronoi cells (not to be confused with the original Voronoi set) with volume $V_i^\text{ico}$ of atom index $i$.
We further employed the other structure characterization methods based on the radial distribution function $g(r)$ and strain localization width $w_\text{sb}$.
Finally, the glass quality, as a mechanical property, was gauged based on the softening modulus
\begin{equation}
h_\text{min} = -\partial_\gamma \sigma_{xy}\Big|_{\rm min}
\end{equation}
denoting the rate of stress decay associated with the maximum instability state upon failure (see Fig.~\ref{fig:hminIllustration}).
In the following subsections, we probe inter-correlations between the above metrics and outline commonalities and differences based on our ``good-versus-bad" theme.

\begin{figure}
    \raggedright
    \begin{overpic}[width=0.27\textwidth]{Figs/scatterD2minRhoGood.png}

        \Labelxy{-8}{35}{90}{$D^2_\text{min}$ (\r{A}$^2$)}
        \Labelxy{17}{15}{0}{$c_{XY}=-0.4$}
         \LabelFig{17}{80}{$(a)$ \tiny Co$_5$Cr$_2$Fe$_{40}$Mn$_{27}$Ni$_{26}$}
         \put(116,10){\includegraphics[height=0.15cm,width=1.7cm]{Figs/colorBar_new.png}}
         \put(116,4) {$\footnotesize \text{Low~~~~~~~High}$}
         \put(106,19){\includegraphics[width=0.15\textwidth]{Figs/d2minGood2000000.png}}
    \end{overpic}
    %

     \begin{overpic}[width=0.27\textwidth]{Figs/scatterD2minRhoBad.png}
        \Labelxy{38}{-3}{0}{$\rho_\text{ico}$(\r{A}$^{-3}$)}
        \Labelxy{-8}{35}{90}{$D^2_\text{min}$ (\r{A}$^2$)}
        \Labelxy{17}{15}{0}{$c_{XY}=-0.2$}
        \put(106,19){\includegraphics[width=0.15\textwidth]{Figs/d2minBad2000000.png}}
        \LabelFig{17}{80}{$(b)$ \tiny CoNiCrFe}
        \begin{tikzpicture}
             \coordinate (a) at (0,0);
             \node[white] at (a) {\tiny.};

    		 \scaleBarr{6.2}{0.0}{0.35}{0.17}{$\footnotesize 0$}{$\footnotesize 20$}{$\footnotesize 40$}{\r{A}}

             \coordSysTwoDim{4.85}{0}{.55}{1}{0}{0}{1}
        \end{tikzpicture}
    \end{overpic}
	%
    \caption{Linking local icosahedral ordering with $D^2_\text{min}$ in the \textbf{a}) Co$_5$Cr$_2$Fe$_{40}$Mn$_{27}$Ni$_{26}$ and \textbf{b}) CoNiCrFe glasses at $\gamma=0.2$. The left panels display scatter plots of $D^2_\text{min}$ and icosahedra density $\rho_\text{ico}$ with the (red) symbols denoting binning averaged data (in log space). The corresponding $D^2_\text{min}$ maps are shown on the right with the blue and red colors indicating regions with low and high nonaffine displacements. The (yellow) dots denote atoms with full icosahedral order.}
    \label{fig:scatterD2minRho}
\end{figure}

\subsection{Structural \& Dynamical Descriptors}
The color maps in Fig.~\ref{fig:scatterD2minRho} overlay atoms with the full icosahedral order (yellow disks) on the two-dimensional (interpolated) $D^2_\text{min}$ field associated with the Co$_5$Cr$_2$Fe$_{40}$Mn$_{27}$Ni$_{26}$ (top) and CoNiCrFe (bottom) metallic glasses.
It is very clear on both maps that (red) rearranging regions significantly lack local structural ordering as opposed to the (blue) rigid matrix.
As a metric to quantify this observation, a cross correlation analysis was carried out between the squared nonaffine displacements of the yellow discs and the associated (number) density $\rho_\text{ico}$.
The scatter data of $D^2_\text{min}$ and $\rho_\text{ico}$ are shown in Fig.~\ref{fig:scatterD2minRho}(a, b) along with the binning-averaged data (in the logarithmic space).
The observed trend indicates significant (anti-)correlations between the logarithm of the two observables $X=\text{log}_{10}{D^2_\text{min}}$ and $Y=\text{log}_{10}\rho_\text{ico}$ with the (linear) correlation coefficient $c_{XY}\simeq -0.4, -0.2$ associated with Co$_5$Cr$_2$Fe$_{40}$Mn$_{27}$Ni$_{26}$ and CoNiCrFe, respectively.
Here $c_{XY}=\langle~\hat{X}_{i}.\hat{Y}_{i}\rangle$ where $\langle.\rangle_i$ denotes averaging over the atom index $i$ and $\hat{X}$ indicates the fluctuating part (with the mean value subtracted) normalized by the standard deviation associated with each variable.

Stronger anti-correlations associated with the ``good" glass imply an infrequent occurrence of the structural ordering within the deforming bands potentially indicative of a strain-softening mechanism.
The ``bad" glass, by contrast, favors a spontaneous formation of (relatively) denser icosahedral clusters inside shear zones with partial strengthening effects that may account for the shallow stress decay associated with its bulk response.
Variations in the relative fraction of ordered clusters is also manifested in the corresponding $D^2_\text{min}$ probablity distribution function.
As shown in Fig.~\ref{fig:pdfCondD2min}, both glasses exhibit a clear bimodal behavior with the first higher and second lower peaks denoting the population of icosahedral clusters forming outside and within  shear zones, respectively.
Results of the Gaussian mixture model associated with the (bad) CoNiCrFe glass (the red squares) indicates a relatively higher contribution of the ordered icosahedral phase to the plastic flow in comparison with the (good) Co$_5$Cr$_2$Fe$_{40}$Mn$_{27}$Ni$_{26}$ metallic glass (the black circles).
This observation is also in line with the outcomes of our correlation analysis.

Our analysis suggests that the interplay between the shear zones dynamics and nucleation of ordered icosahedral clusters might indeed control the extent of shear banding instability in metallic glasses.
In this framework, we probed the evolution of such correlations with the applied strain as well as their variation with glass compositions.
Figure~\ref{fig:cxyGamma}(a) illustrates that the degree of (anti-)correlations between $D^2_\text{min}$ and $\rho_\text{ico}$ grows monotonically with strain by the time it saturates at $\gamma\simeq0.2$ but also reveals meaningful variations with the glass quality.
As shown in Fig.~\ref{fig:cxyGamma}(b), the latter, quantified by $h_\text{min}$, strongly correlates with the ultimate correlation value where better glasses (with lower $h_\text{min}$) correspond, on average, to a lower $c^\infty_{XY}$.

\begin{figure}[t]
    \begin{overpic}[width=0.27\textwidth]{Figs/pdfCondD2minGoodBad.png}
        \Labelxy{-8}{27}{90}{$P(\text{Log}_{10} D^2_\text{min})$}
        \Labelxy{38}{-3}{0}{$\text{Log}_{10} D^2_\text{min}$}
        \begin{tikzpicture}
            \coordinate (a) at (0,0);
            \node[white] at (a) {\tiny.};

			\coordinate (center2) at (2.55,1.0);
            \coordinate (b) at ($ (center2) + .1*(1,0) $);
            \coordinate (c) at ($ (center2) - .1*(1,0) $);
            \draw[red][line width=0.2mm] (c) -- (b);
            \coordinate (d) at ($ (center2) - .5*(0,1) $);
            \draw[->,>=stealth,red][line width=0.2mm] (center2) -- (d);
		    \node at ($(center2)+(0.0,0.2)$) {{\color{red}$\scriptstyle \text{Log}_{10} D^2_\text{th}$}}; %
		    \end{tikzpicture}
    \end{overpic}
    \caption{$D^2_\text{min}$ probability distribution function of atoms with the full icosahedral order corresponding to the Co$_5$Cr$_2$Fe$_{40}$Mn$_{27}$Ni$_{26}$ and CoNiCrFe glasses at $\gamma=0.2$. The solid curves indicate results of the Gaussian mixture model. $D^2_\text{th}$ indicated by the arrow separates the two populations. The error bars denote standard errors.}
    \label{fig:pdfCondD2min}
\end{figure}

\begin{figure}[b]
    \raggedright
    \begin{overpic}[width=0.27\textwidth]{Figs/cxyGamma.png}
        \Labelxy{50}{-3}{0}{\large $\gamma$}
        \Labelxy{-8}{42}{90}{\large$c_{XY}$}
         \LabelFig{17}{80}{$(a)$}
         \put(96,9){\includegraphics[width=0.22\textwidth]{Figs/hCxy.png}}
         \LabelFig{110}{74}{$(b)$}
        \Labelxy{175}{34}{90}{ $h_\text{min}$\footnotesize~(Gpa)}
        \Labelxy{132}{-3}{0}{\large$c^\infty_{XY}$}
    \end{overpic}
    %
    \caption{\textbf{a}) Evolution of the correlation coefficient $c_{XY}$ with applied strain $\gamma$ in several metallic glasses. \textbf{b}) Scatter plot of the softening modulus $h_{min}$ and (terminal) correlation value $c^\infty_{XY}$ associated with different glasses. Each symbol on the right denotes a set of different realizations corresponding to a specific glass composition.}
    \label{fig:cxyGamma}
\end{figure}

Another structural metric we probed is the radial distribution function which should allow for discerning local structural differences leading to the observation of fundamentally different dynamics.
As for the bad glasses, we speculate that the favorable icosahedra formation within plastic zones may have a signature on local density fluctuations around highly rearranging atoms.
We make an appropriate conditioning on $g(r)$ by including center atoms $i=1...N_\alpha$ with $D^2_\text{min}>D^2_\text{th}$, i.e.
\begin{equation}
g(r)=\frac{1}{4\pi r^2\Delta rN_\alpha\bar{\rho}}\sum_{i=1}^{N_\alpha}\sum_{j=1}^{N} \delta(r-r_{ij}),
\end{equation}
with the mean number density of the system $\bar{\rho}$, discretization distance $\Delta r$, Kronecker delta $\delta(.)$, and pair-wise distance $r_{ij}$ of atoms $i$ and $j$.
Here the chosen threshold $D^2_\text{th}$ separates the two $D^2_\text{min}$ populations as in Fig.~\ref{fig:pdfCondD2min}.
Figure~\ref{fig:grD2min} plots the conditioned $g(r)$ associated with the Co$_5$Cr$_2$Fe$_{40}$Mn$_{27}$Ni$_{26}$ and CoNiCrFe glasses as well as the difference between that of the latter and the former.
We note that, despite significant differences in icosahedral ordering and dynamic properties, the two metals have almost identical pair distribution functions.
Nevertheless, the first few characteristic peaks in $g(r)$ corresponding to CoNiCrFe are slightly higher than those of Co$_5$Cr$_2$Fe$_{40}$Mn$_{27}$Ni$_{26}$ (see the green diamonds).
This may indicate a relatively stronger presence of short-range ordering within shear spots of the former glass which were shown to be richer (relative to the latter metal) in terms of the density of icosahedral clusters.

\subsection{Shear Band Characterization}

\begin{figure}[t]
    \centering
    \begin{overpic}[width=0.27\textwidth]{Figs/grD2min.png}
        \Labelxy{50}{-3}{0}{\large $r$\normalsize (\r{A})}
        \Labelxy{-8}{42}{90}{\large $g(r)$\normalsize}

		\end{overpic}

    \caption{Pair distribution function $g(r)$ of the Co$_5$Cr$_2$Fe$_{40}$Mn$_{27}$Ni$_{26}$ and CoNiCrFe glasses at $\gamma=0.2$ conditioned based on $D^2_\text{min}>D^2_\text{th}$ (see Fig.~\ref{fig:pdfCondD2min}) associated with center atoms.}
    \label{fig:grD2min}
\end{figure}

Naturally, topological features associated with the shear band nucleation in metallic glasses should have have a strong bearing on the dynamics of stress near the failure point.
Along these lines, we made measurements of shear band widths $w_{sb}$ as a useful metric to quantify the degree of strain localization and sought for potential correlations with the softening modulus $h_\text{min}$.
In the context of an ideal plastic behavior (with $h_\text{min}\simeq 0$), flow is accompanied by a nucleation of multiple small-scale bands that tend to spread uniformly all across the system.
In this case, one may assume that the shear band width approaches the system size, i.e. $w_\text{sb}\simeq L$.
On the other hand, brittle fracture is typically associated with extremely localized and crack-like deformation features often leading to singular stress curve (with $h_\text{min} \rightarrow -\infty$ in the theoretical limit) \cite{ozawa2018random}.
The following aims to explore such an intimate relationship between $w_{sb}$ and $h_\text{min}$ in the context of metallic glasses.

We performed an auto-correlation analysis on local $D^2_\text{min}$ fields to extract relevant micro-structural lengths.
We start by interpolating the squared nonaffine atomic displacements on a fine cubic grid  to obtain the auto-correlation function
\begin{equation}
    c(\vec{r})=\langle~\hat D^2_\text{min}(\vec{r}^{\hspace{2pt}\prime}+\vec{r}).\hat D^2_\text{min}(\vec{r}^{\hspace{2pt}\prime})~\rangle,
\end{equation}
with $\hat{D^2_\text{min}} \doteq (D^2_\text{min}-\langle D^2_\text{min}\rangle)/\text{var}^{\frac{1}{2}}(D^2_\text{min})$.
Figure~\ref{fig:strnCrltn}(a, b) displays the correlation results corresponding to Co$_5$Cr$_2$Fe$_{40}$Mn$_{27}$Ni$_{26}$ and CoNiCrFe glasses at $\gamma=0.2$.
In Fig.~\ref{fig:strnCrltn}(a), $c(r)$ associated with the former alloy shows an early decay regime along the flow direction $x$ before entering a plateau regime at a bigger $r$ due to the shear band formation that spans across the system (see the map of Fig.~\ref{fig:scatterD2minRho}(a)).
\begin{figure}[b]
    \raggedright
    \begin{overpic}[width=0.27\textwidth]{Figs/cr_d2min.2000000.png}
        \LabelFig{-3}{90}{$(a)$ \tiny Co$_5$Cr$_2$Fe$_{40}$Mn$_{27}$Ni$_{26}$}

        \Labelxy{-8}{50}{90}{\large $c(r)$\normalsize}
         \put(100,12){\includegraphics[width=0.18\textwidth]{Figs/cxy_d2min.2000000.png}}
        \begin{tikzpicture}
             \coordinate (a) at (0,0);
             \node[white] at (a) {\tiny.};

     		 \scaleBarr{6.0}{0.0}{0.35}{0.17}{$\footnotesize 0$}{$\footnotesize 20$}{$\footnotesize 40$}{\r{A}}

             \coordSysTwoDim{6.35}{1.95}{.55}{1}{0}{0}{1}
        \end{tikzpicture}

    		 \end{overpic}
    \begin{overpic}[width=0.27\textwidth]{Figs/cr_d2minBad.2000000.png}
        \LabelFig{-3}{90}{$(b)$ \tiny CoNiCrFe}
        \Labelxy{47}{-3}{0}{\large $r$\normalsize (\r{A})}
        \Labelxy{-8}{45}{90}{\large $c(r)$\normalsize}
         \put(100,12){\includegraphics[width=0.18\textwidth]{Figs/cxy_d2minBad.2000000.png}}
         \put(116,6){\includegraphics[height=0.15cm,width=1.7cm]{Figs/colorBar_new.png}}
         \put(116,0) {$\footnotesize \text{Low~~~~~~~High}$}
         \begin{tikzpicture}
             \coordinate (a) at (0,0);
             \node[white] at (a) {\tiny.};

             \coordSysTwoDim{6.35}{2.25}{.55}{1}{0}{0}{1}
        \end{tikzpicture}
    \end{overpic}

    \caption{$D^2_\text{min}$ correlations corresponding to the \textbf{a}) Co$_5$Cr$_2$Fe$_{40}$Mn$_{27}$Ni$_{26}$ and \textbf{b}) CoNiCrFe glasses at $\gamma=0.2$.  The correlation function $c(r)$ is plotted along the flow direction $x$ and gradient direction $y$. The correlation maps $c(x,y,z=0)$ are shown on the right. The flat (dashdotted) line indicate zero correlations. }
    \label{fig:strnCrltn}
\end{figure}

Along the gradient direction $y$, however, $c(r)$ exhibits a nonmonotonic behavior; it crosses zero at $r\simeq10$~\r{A} and reaches a minimum at a larger length implying the presence of strong anti-correlations.
The former length should give a reasonable width estimate for the nucleated bands, in visual agreements with the $D^2_\text{min}$ map of Fig.~\ref{fig:scatterD2minRho}(a).
Negative correlations may suggest a ``coexistence" between intensely deforming plastic zones and the surrounding solid-like matrix with negligible shear deformation.
The auto-correlation function becomes positive at $r>35$~\r{A} owing to the spontaneous formation of two shearing bands which are roughly apart by the indicated distance.
The CoNiCrFe glass, on the other hand, shows monotonic slowly-decaying correlations in both $x$ and $y$ directions hitting the noise floor at $r\simeq40$~\r{A} which is roughly half the physical system size.
The picture is fairly consistent with the spatial structure of plastic zones with a fairly homogeneous distribution of nonaffine deformations across the bulk, as in Fig.~\ref{fig:scatterD2minRho}(b).

Figure~\ref{fig:SoftModuAndWidth} plots the $w_\text{sb}$ and $h_\text{min}$ estimates for different glasses against the corresponding atomic size mismatch parameter $\delta_a = \sqrt{\text{var}(a_i)}/\langle a_i\rangle$.
The latter is defined as the normalized standard deviation associated with radius $a_i$ of atom $i=1...N$.
In the context of high-entropy alloys, $\delta_a$ is believed to strongly control the degree of local lattice distortions \cite{shang2021mechanical}.
We generalize this concept to glassy alloys where the size mismatch parameter may be thought as a structural proxy controlling the degree of quenched disorder.
Here $\delta_a$ varies between $1-2\%$.
The scatter plot of the softening modulus $h_\text{min}$ and size mismatch $\delta_a$ in Fig.~\ref{fig:SoftModuAndWidth}(a) indicates a positive correlation between the two observables although the CoCrFeMn samples do not seem to follow the overall trend.
Likewise, the shear band width $w_\text{sb}$ in Fig.~\ref{fig:SoftModuAndWidth}(b) seem to be statistically related with $\delta_a$.
Our data (not shown) also suggest that the glass goodness, featured by $h_\text{min}$, correlates with the localization degree measured by $w_\text{sb}$.

\begin{figure}[t]
    \centering
    %
    \begin{overpic}[width=0.23\textwidth]{Figs/SoftModuAndDelta.png}
        \LabelFig{17}{80}{$(a)$}
        \Labelxy{50}{-5}{0}{\large $\delta_a$}
        \Labelxy{-8}{26}{90}{\large $h_\text{min}$\normalsize (Gpa)}
    \end{overpic}
    %
    \begin{overpic}[width=0.23\textwidth]{Figs/WidthAndDelta.png}
        \LabelFig{17}{80}{$(b)$}
        \Labelxy{50}{-5}{0}{\large $\delta_a$}
        \Labelxy{97}{32}{90}{\large $w_\text{sb}$\normalsize (\r{A})}
    \end{overpic}
    %
    \caption{Scatter plot of \textbf{a}) softening modulus $h_\text{min}$ and \textbf{b}) shear band width $w_\text{sb}$ plotted against the size mismatch parameter $\delta_a$ corresponding to the FeNi(\protect\circTxtFill{0}{0}{white}), CoNiFe(\protect\legSqTxt{0}{0}{red}), CoNiCrFe(\protect\legDiamondTxt{0}{0}{darkspringgreen}), CoCrFeMn(\protect\legTriangleTxt{0}{0}{blue}), CoNiCrFeMn(\protect\leftTriang{0}{0}{cyan}), Co$_5$Cr$_2$Fe$_{40}$Mn$_{27}$Ni$_{26}$ (\protect\rightTriang{0}{0}{brown}) metallic glasses.}
    \label{fig:SoftModuAndWidth}
\end{figure}

\section{Discussions}\label{sec:discussions}
Our atomistic simulations of sheared metallic glasses with several principal components have led to the observation of different strain patterns, ranging from a homogeneous plastic-like flow to shear band dominated localized deformation, in agreement with previous numerical and experimental studies.
We have found a direct relevance of underlying microstructure on the atomic-scale dynamics which strongly correlates with the observed deformation features as well as the bulk mechanical response upon failure.
Our Voronoi-based analysis demonstrates the prevalence of the short-range order based on an icosahedral symmetry attributed to the local inter-atomic topology.
Atoms with the observed symmetry are found to be less prone to undergo intense shear transformations.
This has been validated by observing pronounced anti-correlation features between (atomic-scale) nonaffine displacements and density of icosahedral clusters.
Our correlation analysis is similar in essence to the methodology used in \cite{fan2020machine,wang2019transferable} connecting local structure and dynamics in several glass forming metals.
This provides a useful metric to measure structural (in)homogeneities within shear bands which also strongly couple to the glass ability to (de)localize strain.
The extent of bi-modality in the nonaffine displacement distributions is another signature of inherent heterogeneities in a glassy metal that may limit shear strain localization.
To our knowledge, this bi-modal behavior had not been previously reported in the literature.
Based on the radial distribution function data, sheared samples with uniform deformation patterns tend to have (relatively) abundant solid-like structures within flowing regions which may be also understood in terms of the heterogeneity notion.
The latter was further quantified topologically through the (effective) band width measurements which show meaningful variations with the rate of stress decay (as a bulk mechanical property) as well as intrinsic heterogeneities associated with the constituent elements' radii.

Figure~\ref{fig:barchart} provides a summary of key compositional/structural/mechanical metrics probed in this study.
The bar charts sort the element compositions for each index based on the ascending order in the softening modulus $h_\text{min}$ \add{(with the lowest $h_\text{min}$ marking the best glass)}.
Based on the observed trends in the data, $h_{\rm min}$ qualitatively correlates with $\delta_a$ but also, with the shear band width $w_\text{sb}$ and structure-dynamic correlations $c^{\infty}_{XY}$.
\add{One may, however, infer stronger (positive) correlations between the studied order parameters within the scatter data as in Fig.~\ref{fig:cxyGamma}(b) and Fig.~\ref{fig:SoftModuAndWidth} owing to the fact that the bar charts illustrate \emph{mean} quantities associated with $h_\text{min}$, $w_\text{sb}$, and $c^{\infty}_{XY}$ over different realizations}.
\add{Our data indicate that good glasses ---with lowest $h_\text{min}$, $w_\text{sb}$, and $c^{\infty}_{XY}$--- statistically correlate with low misfits, opposite to the commonly-observed trend that the glass forming ability (in the thermodynamics sense) tends to improve with increasing $\delta_a$.
In other words, a glass that shears well is not a good glass former!
}

The observed correlations highlight the significance of chemical compositions and demand for a more systematic search strategy over a much broader composition space to design desired glasses.
It is, therefore, essential to seek for a more descriptive and comprehensive set of relevant structural/physical/mechanical features and probe potential correlations with specific material properties, a training task amenable to materials informatics~\cite{frydrych2021materials}.

\begin{figure}[b]
    \centering
    %
    \begin{overpic}[width=0.5\textwidth]{Figs/barchart.png}
        \Labelxy{22}{0}{0}{\large $h_\text{min}$}
        \Labelxy{42}{0}{0}{\large $\delta_a$}
        \Labelxy{62}{0}{0}{\large $w_\text{sb}$}
        \Labelxy{82}{0}{0}{\large $c^\infty_{XY}$}
        \Labelxy{0}{15}{90}{Scaled Quantities}
    \end{overpic}
    %
    \caption{Softening modulus $h_\text{min}$, misfit parameter $\delta_a$, shear band width $w_\text{sb}$, and structure-dynamics correlations $c^\infty_{XY}$ corresponding to the FeNi(\protect\legRectTxt{0}{0}{black}), CoNiFe(\protect\legRectTxt{0}{0}{red}), CoNiCrFe(\protect\legRectTxt{0}{0}{darkspringgreen}), CoCrFeMn(\protect\legRectTxt{0}{0}{blue}), CoNiCrFeMn(\protect\legRectTxt{0}{0}{cyan}), Co$_5$Cr$_2$Fe$_{40}$Mn$_{27}$Ni$_{26}$ (\protect\legRectTxt{0}{0}{brown}) metallic glasses.}
    \label{fig:barchart}
\end{figure}

The microstructural heterogeneity highlighted in our study must have a strong bearing on local elasticity and associated fluctuations in space \cite{wang2018spatial}.
The latter is a key element in STZ-based mesoscopic models which typically make no direct reference to atomic-scale disorder but instead take into consideration elastic heterogeneity as a main (often phenomenological) model ingredient.
These meso-scale treatments, however, largely ignore true microstructural origins associated with elastic inhomogeneity above some certain scales, which in turn impair predictive capacities of coarse-grained approaches.
In this framework, a direct structural characterization of metallic glasses (via simulations and/or experiments) seems necessary in that local atomic configurations can provide insights on the nature of elastic heterogeneity and its multi scale notion in glassy materials.
These efforts should lead to significant improvements in existing coarse-grained models without compromising the true underlying physics.

\section{Conclusions}\label{sec:conclusions}
In summary, we have identified relevant structural features associated with differing degrees of strain localization observed in mechanically deforming model metallic glasses.
We have based our approach on finding meaningful links between the micorstructure and atomic-scale dynamics in response to an external (shear) stress.
In particular, we established that \emph{\romannum{1}}) structural heterogeneity controls the glass quality described based on the degree of strain localization \emph{\romannum{2}}) the emerging short range order leads to a low susceptibility against localized rearrangements in good glasses enhancing their shear band instability \emph{\romannum{3}}) an abundance of short range ordering within shear zones in bad glasses increases structural inhomogeneity and local hardening and, therefore, their ability to delocalize strain.
Our results have important implications in terms of micro-structural tuning to optimize specific glass properties.


\section{Introduction}
Coherent phonons are widely used to study phonon dynamics for a wide variety of materials such as semimetals,\cite{Cheng1991, Zeiger1992,DeCamp2001, Ishioka2006,Katsuki2013}
semiconductors,\cite{Cho1990,Dekorsy1993,Merlin1996, Hase2003,Hu2011,Mizoguchi2013, Shimada2014, Hayashi2014} superconductors,\cite{Chwalek1991, Albrecht1991, Misochko2000, Takahashi2011} and topological insulators,\cite{Wu2008, Kamaraju2010,Norimatsu2014, Misochko2015} with pump-probe type time-resolved reflectivity measurements. An ultrashort pump pulse coherently excites optical phonons which oscillate in phase and modulate the electric susceptibility. The probe pulse monitors this modulation via a change in the reflectivity.\cite{Dekorsy2000} Thus we directly measured the time evolution of the optical phonons using the pump-probe experiment. Note that we cannot measure the time evolution using conventional frequency-domain spectroscopy because of the time resolution.

Coherent optical phonons are generated by an ultrashort pulse via photon-electron and electron-phonon coupling. The well known mechanisms for coherent phonon
generation are impulsive stimulated Raman scattering (\ISRS)~\cite{Nelson1985} and impulsive absorption (\IA) under the displacement potential~\cite{Zeiger1992, comment} for transparent and opaque regions, respectively. However, past theoretical analyses~\cite{Kuznetsov1994, Merlin1997, Garcia2004, Shinohara2010, Shimada2012} deal with the two processes separately and show that the phonon oscillation can be fitted using sine and cosine functions for the \ISRSs and \IA, respectively. Also, several experiments showed that the phonon oscillation often shifted from the sine or cosine oscillations and its phase was dependent on the materials and phonon modes.\cite{Stevens2002, Ishioka2006, Norimatsu2014} Thus, the generation mechanism for coherent phonons is still controversial, especially for the opaque region where both the light absorption and Raman processes coexist.

The objective of this paper is to determine the generation mechanism of coherent optical phonons by evaluating the contribution of both \ISRSs and \IAs based on a simple quantum-mechanical model. We derive the time evolution of the electron-phonon coupled state by solving the time-dependent Schr\"odinger equation with density matrices using a two-level model for both processes. The quantum-mechanical calculation of the phonon dynamics in the weak coupling limit of the electron-phonon coupling makes it possible to compare the amplitudes and phases of the phonon oscillation for each process quantitatively as a function of the pulse width and the detuning. It was shown that under the resonant condition the \IAs process is dominant for coherent phonon generation and the oscillation can be described by a cosine function. When the optical pulse width increases, the contribution of the \ISRSs process increases and the phonon oscillation deviates from the cosine function. Also, the initial phase of the coherent phonons changes at a long pulse excitation because the contribution of the \ISRSs increases. As the detuning of the excitation wavelength becomes large, the amplitude of the coherent phonons in the \IAs process decreases rapidly, and the \ISRSs process becomes dominant.

\section{Model and formulation}
We consider a two-level system for the electronic state and a harmonic oscillator for the optical phonon at $\Gamma$-point ($\vk=0$). Although this is a crude model for bulk semiconductors, it describes the essential features of the generation of coherent phonons.\cite{Hayashi2014}
It was remarked that this model can be applied to molecular vibrational spectroscopy.\cite{Pollard1990,Banin1994,Cerullo2008} The creation and annihilation operators of the LO phonon at the $\Gamma$-point with energy $\hbar\omega$ are denoted by $b^\dagger$ and $b$, respectively. It was assumed that the excited states are coupled with the LO phonon mode through the deformation potential interaction with the dimensionless coupling constant $\alpha (>0) $. We have approximated the interactions by neglecting the $\vk$-dependence of the coupling constant, and have assumed that the rigid-band shift is because of the deformation potential interaction. In the bulk crystal, the Huang-Rhys factor $\alpha^2$ is considered to be small, $\alpha^2 \ll 1$. The Hamiltonian, $H_0$, is given by
\begin{eqnarray}
H_0 &= &H_g | g \rangle \langle g | + (\epsilon + H_e )  | e \rangle \langle e | \\
H_g &= &\hbar \omega b^\dag b  \\
H_e &= &\hbar \omega b^\dag b + \alpha \hbar \omega (b^\dag +b), \label{dph}
\end{eqnarray}
where $H_g$ and $H_e$ are phonon Hamiltonians for the electronic ground $ | g \rangle$ and excited $ | e \rangle$  states, respectively.
Using the rotating wave approximation, the interaction between the pump pulse and the electronic state is given by
\begin{eqnarray}
H_I(t) = \mu E_0 f(t) \left( e^{-i\Omega t}   | e \rangle \langle g | +  e^{i\Omega t}
| g \rangle \langle e | \right),
\end{eqnarray}
where $\mu$ is the transition dipole moment, $\Omega$ is the central frequency of the pulse, and
$E_0f(t)$ is the envelope of the pulse.~\cite{fcomment} It is noted that these conditions are for a Fourier-transform-limited pulse.

The time evolution of the electron-phonon coupled state was obtained by solving the time-dependent Schr\"odinger equation:
\begin{equation}
i\hbar\frac{d}{dt} |\psi(t)\rangle =\{H_0 + H_I(t)\} |\psi(t)\rangle,
\end{equation}
which gives
\begin{widetext}
\begin{equation}
 | \psi (t)  \rangle = \exp\left(-  \frac{i}{\hbar } H_0 t \right)
\exp_{+} \left( -\frac{i}{\hbar} \int_{-\infty}^t \tilde{H}_I (t^{\prime}) dt^{\prime}\right)  | \psi (-\infty)  \rangle
\end{equation}
with
\begin{equation}
 \tilde{H}_I (t^{\prime}) = \exp\left(\frac{i}{\hbar } H_0 t^{\prime} \right) H_I(t^{\prime})  \exp\left(- \frac{i}{\hbar } H_0 t^{\prime} \right) ,
\end{equation}
where $ | \psi (-\infty)  \rangle$ is the wave function for the initial state at $t=-\infty$ and
$\exp_{+}$ is the time-ordered exponential. Here, $\tilde{H}_I (t^{\prime})$ is given by
\begin{equation}
 \tilde{H}_I (t) = \mu E_0 f(t) \exp\left[ \frac {i}{\hbar}(\epsilon - \hbar  \Omega + H_e) t \right]
  | e \rangle \langle g | \exp\left(-\frac{i }{\hbar}H_g t\right)+ c.c..
\end{equation}
The density matrix of the electronic and phonon states is given by $\rho (t) \equiv  | \psi (t)  \rangle  \langle  \psi (t) |$.
The $ \mu^2$ term of the density matrix $\rho ^{(2)}(t)$ is
\begin{eqnarray}
\rho ^{(2)} (t) &=&  \left(\frac{\mu E_0}{\hbar}\right)^2\exp\left(-  \frac{i}{\hbar } H_0 t \right) F(t)
| g,0  \rangle \langle g, 0 |  F^{\dag}(t) \exp\left(  \frac{i}{\hbar } H_0 t \right)\nonumber \\
&&-\left(\frac{\mu E_0}{\hbar}\right)^2\exp\left(-  \frac{i}{\hbar } H_0 t \right)
 G(t)   | g,0  \rangle  \langle g, 0 | \exp\left(  \frac{i}{\hbar } H_0 t \right)\nonumber \\
&&- \left(\frac{\mu E_0}{\hbar}\right)^2\exp\left(-  \frac{i}{\hbar } H_0 t \right)   | g,0  \rangle
\langle g, 0 |   G^{\dag}(t) \exp\left(  \frac{i}{\hbar } H_0 t \right),
\label{density}
\end{eqnarray}
where the initial state is the electronic ground state and the zero phonon state $| g,0  \rangle$,
and $F(t)$ and $ G(t)$ are given by
\begin{eqnarray}
F(t) &=& \int_{-\infty}^t dt^{\prime} f(t^{\prime}) A^{\dag} (t^{\prime})B(t^{\prime}) | e \rangle \langle g |, \\
G(t) &=&  \int_{-\infty}^{t} dt^{\prime}\int_{-\infty}^{t^{\prime}}  dt^{\prime\prime} f(t^{\prime})   f(t^{\prime\prime}) B^{\dag} (t^{\prime}) A (t^{\prime}-t^{\prime\prime})
B(t^{\prime\prime})  | g \rangle \langle g | .
\end{eqnarray}
\end{widetext}
Both $A(t)$ and $B(t)$ are phonon propagators given by
\begin{eqnarray}
A(t) &=& \exp\left[ -\frac{i}{\hbar} (\epsilon - \hbar  \Omega + H_e) t \right] ,\\
B(t) &=& \exp\left( -\frac{i}{\hbar}  H_g t\right).
\end{eqnarray}
Each term in the right hand side of Eq. (\ref{density}) corresponds to the process described by
the double-sided Feynman diagrams (Fig. \ref{Feynman} (a -- c)).
The first term corresponds to \IAs (Fig. \ref{Feynman} (a)) and the second and third terms correspond to the \ISRS. Hereafter, each term on the right side of Eq. (\ref{density}) for
$\rho ^{(2)} (t) $ is abbreviated to $\rho ^{(2)}_a (t) $, $\rho ^{(2)}_b (t) $, and $\rho ^{(2)}_c (t) $,
respectively. It should be emphasized that these three processes occur equivalently in quantum
mechanics but have different signs in three terms in Eq. (\ref{density}).
\begin{figure}
\begin{center}
\includegraphics[width=6.5 cm]{F-diagram.eps}
\caption{Double-sided Feynman diagrams representing the photo-induced processes: (a) shows \IA, and (b) and (c) show \ISRS. The left and right lines represent the ket and the bra, respectively. The wavy lines represent the photons. Time runs vertically from the bottom to the top.}
\label{Feynman}
\end{center}
\end{figure}

The density matrix $\rho^{(2)}_a (t)$ has only the component $ \langle e | \rho^{(2)}_a (t) | e \rangle$,
which corresponds to the \IAs process, where the phonon oscillates on the adiabatic potential of
the excited state after optical absorption. However, the density matrix $\rho^{(2)}_b (t)$ has
only a nonzero matrix element, $ \langle g| \rho^{(2)}_b (t) | g \rangle$, and the coherent phonons
are generated in the electronic ground state. The density matrix $ \rho^{(2)}_c (t) $ is the Hermitian conjugate to $ \rho^{(2)}_b (t)$.
These two processes represent the \ISRS, in which the phonon oscillates on the ground state adiabatic potential.
The coherent phonon dynamics can be investigated by calculating the mean value of the phonon coordinate $\langle Q (t) \rangle = \mathrm{Tr}\{ Q \rho ^{(2)} (t) \}$,
where $Q \equiv \sqrt{\hbar/2\omega}\left(b+b^\dag\right)$ and $\mathrm{Tr}$ indicates that the trace should be taken over the electronic and phonon variables.
The displacement of each process is separated as follows:
\begin{equation}
\langle Q(t)\rangle =\langle Q_A(t) \rangle +\langle Q_R(t) \rangle,
\end{equation}
where
\begin{equation}
\langle Q_A(t) \rangle=\mathrm{Tr} \{Q \rho ^{(2)}_a (t) \} ,
\end{equation}
and
\begin{equation}
\langle Q_R(t) \rangle=\mathrm{Tr} \{Q \rho ^{(2)}_b (t) \} + \mathrm{Tr} \{Q\rho ^{(2)}_c (t)\}.
\end{equation}

The expressions for the density matrices are easily calculated in our model. For the \IAs process,
we find
\begin{equation}
\langle e| \rho^{(2)}_a (t) | e \rangle =\left(\frac{\mu E_0}{\hbar}\right)^2 |\varphi (t)\rangle
\langle \varphi (t) |,
\end{equation}
where $|\varphi(t) \rangle $ is the phonon wave function given by
\begin{widetext}
\begin{eqnarray}
|\varphi(t) \rangle &=& \int_{-\infty}^t dt^{\prime} f(t^{\prime}) \exp\left[ -\frac{i}{\hbar}
\left(\epsilon -\hbar\Omega+H_e\right)(t-t^{\prime})\right] |0\rangle \nonumber\\
&=& \int_{-\infty}^t dt^{\prime} f(t^{\prime})  U^\dag (\alpha) |\alpha e^{-i\omega(t-t^{\prime})}\rangle
e^{-\frac{i}{\hbar}(\epsilon -\hbar\Omega-\alpha^2 \hbar\omega )
(t-t^{\prime})}.
\end{eqnarray}
Here, the shift operator $U(\beta)$ is defined for an
arbitrary complex parameter $\beta$ by
$U(\beta) \equiv \exp[\beta b^\dag -\beta^* b]$ and the coherent state $|\beta \rangle$ is given by $|\beta\rangle =U(\beta)|0\rangle $.
Then, we obtain
\begin{eqnarray}
\langle Q_A(t)\rangle &=& \alpha\left(\frac{\mu E_0}{\hbar}\right)^2  \sqrt{\frac{\hbar}{2\omega}}
\int_{-\infty}^t dt^{\prime}\int_{-\infty}^t  dt^{\prime\prime} f(t^{\prime})f(t^{\prime\prime}) \left(e^{-i\omega(t-t^{\prime})}+ e^{i\omega(t-t^{\prime\prime})}-2\right) \nonumber \\
& & \times\exp\left[- \alpha^2 \{1+i\omega (t^{\prime}-t^{\prime\prime})-e^{i\omega(t^{\prime}-t^{\prime\prime})}\}\right] e^{i(\epsilon -\hbar\Omega)(t^{\prime}-t^{\prime\prime})/\hbar}.
\label{QAgeneral}
\end{eqnarray}

For the \ISRS, we can write
\begin{equation}
\langle g| \rho^{(2)}_b (t) | g \rangle =-\left(\frac{\mu E_0}{\hbar}\right)^2 |\chi(t)\rangle
\langle 0 |,
\end{equation}
where $|\chi(t) \rangle $ is given by
\begin{eqnarray}
|\chi(t) \rangle &=& \int_{-\infty}^t dt^{\prime}\int_{-\infty}^{t^{\prime} } dt^{\prime\prime} f(t^{\prime}) f(t^{\prime\prime})
\exp\left[ -\frac{i}{\hbar}H_g(t-t^{\prime})\right]
\exp\left[ -\frac{i}{\hbar}
\left(\epsilon -\hbar\Omega+H_e\right)(t^{\prime}-t^{\prime\prime})\right] |0\rangle \nonumber\\
&=& \int_{-\infty}^t dt^{\prime} \int_{-\infty}^{t^{\prime} }dt^{\prime\prime} f(t^{\prime}) f(t^{\prime\prime}) U^\dag (\alpha e^{-i\omega (t-t^{\prime})})
|\alpha e^{-i\omega(t-t^{\prime\prime})}\rangle e^{-i(\epsilon -\Omega -\alpha^2
\hbar\omega)(t^{\prime}-t^{\prime\prime})/\hbar}.
\end{eqnarray}
The mean value of $Q_R(t)$ for the \ISRSs process is then given by
\begin{eqnarray}
\langle Q_R(t) \rangle &=& \alpha \left(\frac{\mu E_0}{\hbar}\right)^2
\sqrt{\frac{\hbar}{2\omega}}\int_{-\infty}^t dt^{\prime} \int_{-\infty}^{t^{\prime} }dt^{\prime\prime} f(t^{\prime}) f(t^{\prime\prime})
\left(e^{-i\omega (t-t^{\prime})}-e^{-i\omega (t-t^{\prime\prime})}\right) \nonumber\\
&&\times \exp\left[-\alpha^2\{1-i\omega(t^{\prime}-t^{\prime\prime})-e^{-i\omega(t^{\prime}-t^{\prime\prime})}\}\right]
e^{-i(\epsilon - \hbar\Omega)(t^{\prime}-t^{\prime\prime})/\hbar}+ c.c..
\label{QRgeneral}
\end{eqnarray}
\end{widetext}
Formulae (\ref{QAgeneral}) and (\ref{QRgeneral}) are general expressions for the expectation values of the phonon coordinate in \IAs and \ISRS, respectively.

\section{Results and Discussion}
Let us assume that the pulse-envelope function $f(t)$ is a real quantity
localized in the region $|t|\lesssim\sigma$ around $t=0$ with the normalization condition
$\int_{-\infty}^\infty f(t) dt  =1$. As a typical example, we set the Gaussian function to be
\begin{equation}
f(t)=\frac{1}{\sqrt{\pi}\sigma}\exp(-t^2/\sigma^2).
\label{Gauss}
\end{equation}
In what follows, $(\mu E_0/\hbar)^2 \equiv 1$.
\subsection{Resonant condition}
\begin{figure}[t]
\centering
\includegraphics[width=4.5cm]{Fig2N.eps}
\caption{Schematic illustration of the \IAs and \ISRSs processes on the adiabatic potential curves for the electronic ground and excited states. The horizontal small arrows indicate the direction of the initial motion of the phonon in the electronic excited and ground states.}
\label{APES}
\end{figure}
In this subsection, we focus our attention to the case of resonant excitation with weak
electron-phonon coupling. Setting $\epsilon-\hbar\Omega=0$ and neglecting the terms
of $\alpha^2$ order, we simplify the expressions to give
\begin{eqnarray}
\langle Q_A(t) \rangle &=&\alpha
\sqrt{\frac{\hbar}{2\omega}}\int_{-\infty}^t dt^{\prime} \int_{-\infty}^{t }dt^{\prime \prime}f(t^{\prime}) f(t^{\prime \prime}) \nonumber\\
&&\times\left[ \cos\omega(t-t^{\prime})+\cos\omega(t-t^{\prime \prime})-2 \right],
\label{generalQA}
\end{eqnarray}
\begin{eqnarray}
\langle Q_R(t)\rangle
&=&2\alpha
\sqrt{\frac{\hbar}{2\omega}}\int_{-\infty}^t dt^{\prime} \int_{-\infty}^{t^{\prime}}dt^{\prime \prime}f(t^{\prime}) f(t^{\prime \prime}) \nonumber\\
&&\times
\left[ \cos\omega(t-t^{\prime}) -\cos\omega(t-t^{\prime \prime}) \right].
\label{generalQR}
\end{eqnarray}

We discuss the general features of $\langle Q_A(t)\rangle$ and $\langle Q_R \rangle$ based on the formulae (\ref{generalQA}) and (\ref{generalQR}).
First let us discuss the behavior of $\langle Q_A(t) \rangle $. It is obvious from Eq.(\ref{generalQA}) that $Q_A(t)\leq 0$ since the phonon oscillates on the
excited state potential energy curve as shown schematically in Fig. \ref{APES}.
In the case of the Gaussian pulse (\ref{Gauss}), and for $t \gg \sigma$, the integral is approximately carried out by extending the upper limit to $\infty$, and we find the cosine-like oscillation around
the new equilibrium point $ -Q_0 (\equiv -2\alpha\sqrt{\hbar/2\omega} )$ in the excited state,
\begin{equation}
\langle Q_A(t)\rangle =Q_0 \left(e^{-\sigma^2\omega^2/4}\cos \omega t - 1\right).
\label{Qanalytic}
\end{equation}
In the short pulse limit $\sigma\omega\rightarrow 0$, this reduces to
\begin{equation}
\langle Q_A(t)\rangle =Q_0 \left(\cos \omega t - 1\right),
\end{equation}
which is in agreement with the coherent state. In the long pulse limit,
$\sigma\omega \gg 1$, however, the oscillation disappears and $\langle Q_A(t)\rangle$ changes gradually during the pulse duration from $0$ to the new equilibrium of the lowest vibrational state $\langle Q_0\rangle$ in the excited state. Thus, Eq. (\ref{Qanalytic}) clearly shows the changeover of the phonon dynamics induced by the electronic excitation, from the sudden transition limit $\sigma\omega\rightarrow 0$ to the adiabatic change limit $\sigma\omega\rightarrow \infty$.

In the \ISRSs process described by $\langle Q_R(t)\rangle$, the oscillation
is induced by the sudden occurrence of momentum because of the excitation and the deexcitation
in the impulsive stimulated Raman process. Therefore, its motion is sine-like. In the case of the resonant excitation, $\langle Q_R(t)\rangle$ is calculated as follows.

For the time $t$ after the passage of the optical pulse, $t \gg \sigma$, the time-ordered integral in Eq. (\ref{generalQR}) can be evaluated by extending the upper limit of the integral to infinity:
\begin{widetext}
\begin{equation}
\langle Q_R(t)\rangle=Q_0 e^{-i\omega t}\int_{-\infty}^\infty dt^{\prime} \int_{-\infty}^{t^{\prime} }dt^{\prime \prime} f(t^{\prime})f(t^{\prime \prime})
\left(e^{i\omega t^{\prime}} - e^{i\omega t^{\prime \prime}}\right) + c.c..
\end{equation}
Using new variables $s\equiv(t^{\prime}+t^{\prime \prime})/2$ and  $u\equiv t^{\prime}-t^{\prime \prime}$, we find
\begin{equation}
\langle Q_R(t)\rangle = Q_0 \frac{2 i e^{-i\omega t}}{\pi\sigma^2} \int_{-\infty}^{\infty} ds \int_0^\infty du
\exp\left[ -\frac{2}{\sigma^2}s^2-\frac{1}{2\sigma^2}u^2\right] e^{i\omega s} \sin \left[ \frac{\omega}{2}u \right] +c.c.
= A\sin\omega t.
\label{sine}
\end{equation}
\end{widetext}
The useful formulae~\cite{comment2} are included with the coefficient $A$ given by
\begin{equation}
A \equiv Q_0 \frac{4}{\sqrt{\pi}}e^{-\sigma^2\omega^2/4}\int_0^{\sigma\omega/2\sqrt{2}} e^{t^2} dt.
\end{equation}
Equation (\ref{sine}) tells us that the phonon oscillation is induced by the impulsive generation of momentum because the excitation and the de-excitation occurs through \ISRS. Therefore, its motion is sine-like. This also indicates that the amplitude of the oscillation $A$ takes a maximum value at an intermediate value of the pulse width $\sigma$ because $A\rightarrow 0$ both in the limits $\sigma\rightarrow 0$ ($\delta$-function pulse) and $\sigma\rightarrow \infty$. In other words, it needs a finite duration in the electronic excited state for the phonon to get momentum. Since the coefficient $A$ must be positive, the phonon wave packet begins to move in the direction opposite to $\langle Q_A (t)\rangle$. This counterintuitive phenomenon can be understood as follows.
In the process of \ISRSs described by Fig. \ref{Feynman} (b) and (c), the phonon wave function in the excited state gives a contribution that is the negative of the ground state. This is explicitly shown by the negative signs of the $\rho^{(2)}_b (t)$ and $\rho^{(2)}_c (t)$ in Eq. (\ref{density}). In other words, a \textit{hole} is created in the ground state wave function
of the phonon\cite{Banin1994,Kayanuma2000} by \ISRSs as shown in Fig. \ref{APES} so that the expectation value $\langle Q_A(t)\rangle$ moves in the opposite direction to that in the \IAs process.
\begin{figure}
\centering
\includegraphics[width=7cm]{Fig3N1.eps}
\caption{(Color online) Time evolution of the mean value of the atomic displacement with pulse width (full width at half maximum) of $0.1 T$ (red), $0.55 T$ (green), and $0.9 T$ (blue) for
the vibrational period ($T$) of the phonon: (a) the dotted curves represent $\langle Q_R (t) \rangle$; (b) the solid curves represent $\langle Q_A (t) \rangle$; (c) the solid curves represent $\langle Q (t) \rangle$.}
\label{Q_cal}
\end{figure}

In Figs. \ref{Q_cal} (a) and (b), the numerical results of the dependence of the pulse-width on $\langle Q_A(t)\rangle$ (solid lines) and $\langle Q_R(t)\rangle$ (dashed lines) are shown as a function of the delay-time for the Gaussian pulses. The vibrational period of the coherent phonons was set to be \textit{T}.\cite{comment3} For $\langle Q_A(t)\rangle$, the approximate formula (\ref{Qanalytic}), agrees with the exact results in the region of time after the passage of the pulse. Violent oscillations of the coherent phonons changes to a gradual adaptation of a new equilibrium as the pulse width becomes large.

However, for $\langle Q_R(t)\rangle$, there is an optimum value of the pulse width that maximizes the amplitude of the oscillation. To discriminate experimentally between $\langle Q_A(t)\rangle$ and $\langle Q_R(t)\rangle$, it is necessary to use the electronic state-selective measurement of the coherent phonons. However, in the commonly used transient reflectivity or transmissivity measurements, the electronic states are not identified. One can get information only on the total value $\langle Q(t) \rangle\equiv \langle Q_A(t)\rangle + \langle Q_R(t)\rangle$, which is also shown in Fig. \ref{Q_cal} (c).
$\langle Q(t) \rangle$ shows an oscillation with the frequency $ \omega$, which is almost cosine-like at short pulse widths such as $0.1 T$. As the pulse width increases, the contribution from the \ISRSs process increases and the initial phase of the oscillation changes.

\begin{figure}
\centering
\includegraphics[width=6cm]{Fig4N1.eps}
\caption{(Color online) The dependence of the pulse-width on the amplitude of the oscillation part of the phonons, for $\langle Q_A (t) \rangle$ (red) and $\langle Q_R (t) \rangle$ (blue). \textit{T} is the vibrational period of the phonons.}
\label{amplitude}
\end{figure}
In Fig. \ref{amplitude}, the amplitudes of the oscillation are plotted for \IAs (red line) and \ISRSs (blue line) as a function of the pulse-width (FWHM). The dominant process changes from \IAs to \ISRSs around the pulse width of $0.55 T$. This phenomenon may be experimentally observed from the initial phase of the phonon oscillation by experiments that precisely control the pulse width and frequency chirping.
\subsection{Detuning effect}
The detuning from the resonant condition was studied for $\Delta E \equiv \epsilon - \hbar \Omega > 0$
by evaluating Eqs. (\ref{QAgeneral}) and (\ref{QRgeneral}).
It is worth noting that the $\langle Q_A(t)\rangle$ is also described analytically for $t \gg \sigma$ by
\begin{equation}
\langle Q_A(t)\rangle =Q_0 e^{-\sigma^2 \xi^2/2}  \left(e^{-\sigma^2 (\omega^2+2 \xi \omega )/4}\cos \omega t - 1\right)
\label{Qanalytic2}
\end{equation}
with $\xi \equiv (\epsilon -\hbar \Omega)/\hbar$.

However, $\langle Q_R (t)\rangle$ can be calculated using
\begin{widetext}
\begin{equation}
\langle Q_R (t) \rangle = \frac{1}{\sqrt{2\pi}} e^{-\sigma^2\omega^2/8} e^{-i\omega t}
\int_0^\infty du e^{-u^2/2\sigma^2} \left( e^{i\omega u/2} -e^{-i\omega u/2}\right) e^{-i \xi u} + c.c..
\end{equation}
For general values of the detuning, $\xi$, the above integral yields a complex value, so that the phase of the oscillation of $\langle Q_R (t) \rangle$ changes gradually as the detuning becomes large. We evaluated the initial phase in the limit of large detuning $\xi \sigma \gg 1$ using
\begin{equation}
\int_0^\infty du e^{-u^2/2\sigma^2-i \eta u} = \sqrt{\frac{\pi}{2}}\sigma e^{-\sigma^2 \eta^2/2}-
i\sqrt{2}\sigma e^{-\sigma^2 \eta^2/2}\int_0^{\sigma \eta/\sqrt{2}}e^{t^2} dt
\end{equation}
\end{widetext}
with the positive constant $\eta$.
It is obvious that the real part becomes negligible when compared with the imaginary part in the limit $\sigma \eta \gg 1$. Therefore, in the case of large detuning, we can neglect the real part and obtain
\begin{equation}
\langle Q_R (t)\rangle =B\sin\omega t,
\label{sine2}
\end{equation}
where
\begin{equation}
B \equiv Q_0\frac{2}{\sqrt{\pi}} e^{-\sigma^2\omega^2/8} \left\{D\left(\xi +\frac{\omega}{2}\right)
-D\left(\xi -\frac{\omega}{2}\right)\right\},
\end{equation}
and
\begin{equation}
D(x)\equiv e^{-\sigma^2 x^2/2}\int_0^{\sigma x/\sqrt{2}} e^{t^2} dt.
\end{equation}

In the limit of large detuning, the asymptotic form of $D(x)$ can be obtained from the inverse power-series expansion:
\begin{equation}
D(x)\sim a_1 x^{-1}+a_2 x^{-2} + \cdots.
\end{equation}
Inserting the above expression into the differential equation, $dD(x)/dx=-\sigma^2 x D(x)+\sigma/\sqrt{2}$,
we find to the lowest order term
\begin{equation}
D(x) \simeq \frac{1}{\sqrt{2}\sigma x},
\end{equation}
and
\begin{equation}
B\simeq - Q_0 \sqrt{\frac{2}{\pi}} e^{-\sigma^2\omega^2/8}\frac{\omega}{\sigma \xi^2},
\label{BforlargeD}
\end{equation}
where we have used the approximation
\begin{equation}
D \left( \xi+\frac{\omega}{2} \right) - D \left( \xi-\frac{\omega}{2} \right) \simeq
\frac{dD}{d\xi}\omega.
\end{equation}

\begin{figure}
\centering
\includegraphics[width=7cm]{Fig5N1.eps}
\caption{(Color online) The time evolution of the mean value of the phonon coordinate with detuning $\Delta E$ of $0 \hbar \omega$ (red: no detuning), $3 \hbar \omega$  (green),
$5 \hbar \omega$  (blue), and $10 \hbar \omega$ (purple). (a) The dotted curves represent $\langle Q_R (t) \rangle$, (b) the solid curves represent $\langle Q_A (t) \rangle$ and (c) the solid curves represent $\langle Q (t) \rangle$. $T$ is the vibrational period of the phonons. The pulse width was set to $0.1 T$.}
\label{Q_cal2}
\end{figure}
Therefore, we find $B<0$ in the limit of large detuning. The initial phase of the sine-like oscillation in \ISRSs changes by $\pi$ as the detuning changes from $0$ to large values.
Equation (\ref{BforlargeD}) is the general form for the dependence of the amplitude of the coherent phonon oscillation on the detuning $\xi$ and pulse-width $\sigma$ in the case of pumping in the transparent region.

Figure \ref{Q_cal2} shows the time evolution of the atomic displacements $\langle Q_A(t)\rangle$, $\langle Q_R(t)\rangle$,
and $\langle Q(t)\rangle$ with a pulse width of $0.1 T$. The center position of the atom around which $\langle Q_A(t)\rangle$ oscillates
approaches zero as $\Delta E$ increases. $\langle Q_R(t)\rangle$ starts to move in the same direction as $\langle Q_A(t)\rangle$ at large detuning, as alluded to before.
Figure \ref{amplitude2} shows the amplitude of the oscillation of the phonons. The oscillation amplitude of both $\langle Q_A(t)\rangle$ and $\langle Q_R(t)\rangle$ decreases as $\Delta E$ increases. For large detuning, the oscillation amplitude of $\langle Q_R(t)\rangle$ becomes larger than that of the $\langle Q_A(t)\rangle$ since no light absorption occurs, and
the dominant coherent phonon generation process is subject to the \ISRSs mechanism.
Both of the numerical values of the oscillation amplitude for $\langle Q_A(t)\rangle$ and $\langle Q_R(t)\rangle$ agree with the analytical formulas, namely Eq. (\ref{Qanalytic2}) for $\langle Q_A(t)\rangle$ and Eq. (\ref{sine}) for $\langle Q_R(t)\rangle$ where $\xi=0$ and Eq. (\ref{sine2}) where $\xi \gg \omega$.
\begin{figure}
\centering
\includegraphics[width=6cm]{Fig6N2.eps}
\caption{(Color online) The detuning dependence of the amplitude of the oscillation of the phonons, for $\langle Q_A (t) \rangle$ (red) and $\langle Q_R (t) \rangle$ (blue). The pulse width was set to $0.1 T$.}
\label{amplitude2}
\end{figure}

\section{Conclusion}
In this work, we investigated the generation mechanisms of coherent optical phonons using a simplified two-level model with resonant excitation conditions. The quantum-mechanical calculations indicate that both the optical phonons in the electronic excited $| e\rangle $ and ground $| g\rangle $ states are excited via impulsive absorption and stimulated Raman scattering by irradiation of the femtosecond pulse at resonance. In the short pulse limit, only the optical phonon in the excited state is driven to the coherent state, but the phonons are not excited in the electronic ground state. As the pulse width increased, the amplitude of $| e\rangle $ decreased while that of $| g\rangle $ increased. The mean value of the atomic displacement, $\langle Q_A \rangle$ and $\langle Q_R \rangle $, started to move in opposite directions in $| e\rangle $ and $| g\rangle $, respectively. In the long pulse limit, both of the amplitudes of $\langle Q_A \rangle$ and $\langle Q_R \rangle $ tended to zero. In the intermediate conditions, one should
consider the superposition of \IAs and \ISRSs processes, although \IAs is generally dominant. Our proposed model shows that both well-known coherent phonon mechanisms, \IAs and \ISRS, occur depending on the pulse length and the detuning. The initial phase of the phonon oscillation in the case of resonant excitation changed depending on the pump pulse-width because of competition between the \IAs and \ISRSs mechanisms. This indicates that care must be taken in making generation mechanism arguments based on the initial phase.

In the present work, only the symmetric mode of the phonons was considered. The model can be
easily extended to also include asymmetric Raman modes, which rotate the polarization
of the incident photons.\cite{JJAP} Also, in the usual optical detection of coherent phonons,
atomic displacements are detected indirectly through the modulation of the electric susceptibility.
The extension of the present study to include these processes will be shown in a forthcoming paper.
Further, the present work investigates the coherent control of the lattice vibrations by a pair of optical pulses
with a well-defined phase.~\cite{Katsuki2013,Hayashi2014,Castella1999} The extension of the present study
can be also applied to such double-pulse system.
Finally, the environmental effects on the electronic and the phonon (vibrational) states is to be considered.\cite{Mukamel1995}
It is important to analyze these effects to determine the lifetime and the decoherence mechanism of the coherent phonons.
\begin{acknowledgements}
The authors thank K. Goto of Tokyo Institute of Technology for his effort in the early stage of this work.
This work was partially supported by Core Research for Evolutional Science and Technology (CREST) of the Japan Science and Technology Agency (JST),
JSPS KAKENHI Grant Number 25400330 and 15K13377, the Collaborative Research Project of Materials and Structures Laboratory, NINS Youth Collaborative Project,
and the Joint Studies Program of the Institute for Molecular Science.
\end{acknowledgements}


\section{Experimental details}

Given the broad dynamic range required to probe both the molecular excitations and the lattice modes of the PbBr$_{6}$ host lattice, several instruments with overlapping energy resolutions were required.  We first performed a series of experiments on the IN10 backscattering instrument (ILL, France).  IN10 consists of a large, vibrating Si(111) monochromator that Doppler shifts incident neutrons with energies near E$_{i}$=2.08 meV and directs them onto the sample.  Neutrons scattered from the sample are then energy analyzed by a large bank of Si(111) crystals, which backscatter neutrons having energy E$_{f}$=2.08 meV through the sample and onto a series detectors.  The dynamic range of this spectrometer is $\pm$ 10 $\mu eV$, and the elastic resolution is $\delta E$=0.5 $\mu eV$ (half width).

\renewcommand{\thefigure}{S1}
\begin{figure}[t]
\includegraphics[width=8.5cm] {fixed_window.eps}
\caption{\label{window}  A plot of the integrated intensity of a ``fixed window" scan taken on IN10.}
\end{figure}

Given the limited temperature range where dynamics are observable on IN10 (see Figure 2 of the main text), we then extended these measurements using the NEAT chopper spectrometer (HZB, Berlin).  NEAT is a direct geometry cold chopper spectrometer.  Two different configurations were used with incident wavelengths fixed at 6.3 \AA\ and 12 \AA\ with energy resolutions of $\delta E\sim$ 8 $\mu eV$ and 70  $\mu eV$ respectively.  The results of these measurements are combined with the IN10 data in Fig. 2 of the main text.

A combination of both IN10 and NEAT provide a temperature dependence of the dynamics up to the structural transition from an orthorhombic to tetragonal unit cell.  However, the dynamics were found to quickly leave the energy window at the structural transition at 150 K.  To probe the dynamics over a broader energy range with coarser energy resolution we used the IN4 direct geometry thermal Fermi chopper spectrometer (ILL, France).   The incident energy was fixed to 16.84 meV for the data presented.

To finally probe all of the dynamics as a function of temperature and to provide a dataset amenable for comparisons with calculations (to determine the nature of the soft phonons) we performed a series of experiments on the MARI direct geometry chopper spectrometer at ISIS.  For the data presented in Fig. 4 of the main text, we used a Gd Fermi chopper to fix the incident energy in combination with a disc chopper to remove background from high energies.  These experiments used E$_{i}$=25 meV, 50 meV, and 100 meV spinning the fermi chopper at 400 Hz, 400 Hz, and 600 Hz respectively.    To investigate the internal modes presented here in the supplementary information, the Gd fermi chopper was used with E$_{i}$=150 meV spin at 300 Hz.  For higher energies, the relaxed fermi chopper was used with E$_{i}$=250 meV, 550 meV, and 1000 meV.  The chopper was spun at 300 Hz, 300 Hz, and 600 Hz respectively.   For each spectrometer configuration, we used the low-temperature ($T$=2 K) dataset as the resolution function, and convolved this with the model $S(Q,E)$ after background subtraction.

\section{``Elastic Window" Scan}

\renewcommand{\thefigure}{S2}
\begin{figure}[t]
\includegraphics[width=7.5cm] {Q_depend_fixed_window.eps}
\caption{\label{fixed_q}  A sample plot of the momentum dependence extracted from a ``fixed window" scan done on the IN10 backscattering spectrometer at 145 K and 200 K.  The solid lines are fits to gaussians used to extract $\langle u^{2}\rangle$ as a function of temperature.  The results of this fit are plotted in Fig. 1 of the main text.}
\end{figure}

In the main text, an elastic window scan from the IN10 backscattering spectrometer was used to estimate $\langle u^{2} \rangle$.  In this scan, the instrument integrates over the elastic line and $\pm$ 10 $\mu eV$.  The integrated intensity summed over all detectors is shown in Fig. \ref{window}.  The intensity can be seen to show discontinuities at the same temperatures illustrated in Fig. 1 in the main text.  The decrease in intensity at $\sim$ 50 K does not correspond to any transition in the bulk material and neutron inelastic scattering work (presented in the main text) finds this to be due to methyl group rotations.

We have used this scan to estimate $\langle u^{2} \rangle $ by fitting the momentum dependence at each temperature to the form $I(Q)\sim e^{-\langle u^{2}\rangle Q^{2}}$.  The IN10 backscattering spectrometer consists of seven detectors covering a momentum range from 0.4 \AA$^{-1}$ to 1.96 \AA$^{-1}$ and example fits of the ``fixed window" scan are shown in Fig. \ref{fixed_q}.  The results of these fits at each temperature are shown in Fig. 1 of the main text.

\section{High-energy internal modes}

We now discuss the higher energy lattice modes probed with MARI.  Figure \ref{mari} shows there to be considerable band gap in the lattice excitations between $\sim$ 40 meV and $\sim$ 110 meV.  This gap separates low-energy modes involving the inorganic framework and the molecular excitations and the higher energy internal modes of the MA molecule.   As shown in the main text, while the lower energy band of excitations is highly temperature dependent, the higher energy modes show comparatively little temperature dependence.  Here we use previous Raman spectroscopy results on CsPbCl$_{6}$~\cite{Carabatos03:34} and the antifluorite (MA)$_2$SnCl$_6$~\cite{Shi98:29} to distinguish the two different sets of vibrations and to corroborate the Castep analysis used in the main text to help separate lattice modes from the PbBr$_{6}$ framework and the molecular motions.

\renewcommand{\thefigure}{S3}
\begin{figure}[t]
\includegraphics[width=7.0cm] {mari_castep_2.eps}
\caption{\label{mari}  High energy data on MARI illustrating the two bands of excitations from modes involving the PbBr$_{6}$ framework and the internal molecular motions at high energies.  The two panels illustrate data taken with E$_{i}$=150 meV and 550 meV respectively.  The data is compared to Castep calculations on the left illustrating where a finite neutron cross section is expected.}
\end{figure}

Based on previous studies of (MA)$_{2}$SnCl$_{6}$,~\cite{Shi98:29}, we reproduce the mode assignment presented in Table \ref{table}.   The mode energies are comparable to that observed here for CH$_{3}$NH$_{3}$PbBr$_{3}$ and are corroborated by calculations performed with Castep.

\begin{table}[ht]
\caption{Assignment of the high-frequency modes from the 5~K spectra taken on MARI. Frequencies are from Raman studies on (MA)$_{2}$SnCl$_{6}$ at 8 K.~\cite{Shi98:29}}
\centering
\begin{tabular} {c c c c c }
\hline\hline
$\nu$ & Nature & Energy (meV)\\
\hline\hline
$\nu_{7}$    &  N-H antisymmetric strtech 				& 400  			\\
$\nu_{1}$    &  N-H symmetric stretch 				& 397 			\\
$\nu_{8}$    &  C-H antisymmetric stretch  				& 372 			\\
$\nu_{2}$    &  C-H symmetric stretch 				& 360 			\\
$\nu_{9}$    &  NH$_{3}$ antisymmetric deformation 		& 198 			\\
$\nu_{3}$    &  NH$_{3}$ symmetric deformation 		& 182 			\\
$\nu_{10}$  &  CH$_{3}$ antisymmetric deformation 		& 180 			\\
$\nu_{4}$    &  CH$_{3}$ symmetric deformation 		& 176 			\\
$\nu_{11}$  &  rock								& 155 			\\
$\nu_{5}$    &  C-N stretch 						& 121 			\\
$\nu_{12}$  & rock								& 112			\\
\hline
\label{table}
\end{tabular}
\end{table}

The study of modes involving the lead halide perovskite structure have been performed by several groups including Raman studies in Ref. \onlinecite{Calistru97:82,Carabatos03:34}.  These mode assignments are expected to differ from what is observed here owing to the different between chlorine and bromine.  We also note that Raman spectra in CsPbCl$_{6}$ only shows a comparatively small softening of the low-energy modes in comparison to what we report in the main text for CH$_{3}$NH$_{3}$PbBr$_{3}$.  However, the results qualitatively agree with the energy range of the lattice dynamics that we report in the main text.

\textit{Open data access:} Following UK Research Council guidance, data files can be accessed either at source (from ISIS (www.isis.stfc.ac.uk) or the ILL (www.ill.eu)) or through the University of Edinburgh's online digital repository (datashare.is.ed.ac.uk) after publication.