text:
\section{Introduction}
Commencing in 2004 with the first observation of free standing graphene - along with the measurement of its electronic properties \cite{Geim:graphene2004}- there has been an abundance of both experimental and theoretical work performed on graphene and its multi-layered counterparts \cite{Novo:graphene2005, geim:riseofgraphene2007, nilsson2006electronic, mccann2013electronic, bowlan2014ultrafast, paul2013high}. The impetus behind this research arises from the intriguing electrical, mechanical, and optical properties possessed by graphene. Many of the intriguing properties of monolayer graphene (MLG) are also present in bilayer graphene (BLG). These include a zero-gap energy dispersion, a very high carrier mobility,  high thermal conductivity, and the ability to tune electric properties by adjusting the carrier density through gating \cite{Novo:graphene2005, Morozov:GiantIntrinsic2008}.

Despite their similarities, BLG differs from MLG in a number of ways. One important distinction is found in the density of states of both systems. Close to the \textit{Dirac} points - the points within the first Brillouin zone at which the conduction and valence bands touch - one can show that although the interband matrix elements for both systems are essentially the same, in BLG, the number of states within 10 meV of the Dirac point is approximately 30 times larger than in MLG. This difference in the density of states will have an effect on the interband transitions of the two systems. In order to probe this energy range one needs to probe BLG with radiation that has frequencies in the terahertz range.

As with MLG, the absorption of optical and THz radiation by BLG can be understood in terms of interband and intraband transitions. The energies of photons at THz frequencies are ideally suited to the study of intrinsic (undoped) BLG, where the Fermi level is at the Dirac point. At these frequencies, interband transitions can take place near the Dirac point, while the injected electrons and holes are subsequently strongly driven within their respective bands. As we shall see, the ability of THz radiation to induce both intraband and interband dynamics is fundamental in the emergence of a strong nonlinear response in BLG.

Earlier studies on MLG have suggested the presence of a strong nonlinearity at both optical and THz  frequencies \cite{mikhailov2007non, mikhailovnonlin2009, hendry2010coherent, ishikawa2010nonlinear, wright2009strong}. Theoretical studies of BLG have indicated a strong nonlinear effect in the THz to far-infrared regime, whereby a moderate electric field can result in third harmonic generation at room temperature \cite{ang2010nonlinear}. Experiments have been performed with the intent of observing the nonlinear THz response of graphene, specifically third harmonic generation. Although it has been observed by using a 45-layer sample, it has not yet been successfully observed in MLG or BLG \cite{bowlan2014ultrafast, paul2013high}.

Almost all experimental investigations of the nonlinear response of graphene have observed a response that is dominated by intraband dynamics, due to the high doping level of the samples. The large Fermi energy of the doped system diminishes the interband current due to Pauli blocking. For more moderate Fermi energies - tens of meV - the interband dynamics can make a contribution that is dependent on the field amplitude. In recent theoretical work on MLG, it was shown that if the Fermi level is reduced to within a few meV of the Dirac point, the magnitude of the interband current becomes comparable to the intraband current, resulting in the presence of a strong nonlinearity \cite{al2014high}. Specifically, it was found that odd harmonics of the THz field should be generated. As this interplay between interband and intraband dynamics is a result of the zero-gap band structure, we also expect to see a similar interplay in the carrier dynamics in BLG.

In this paper, we employ a four-band tight-binding method to model the intraband and interband dynamics of undoped suspended bilayer graphene in response to a single-cycle THz pulse in the range of 1 to 5 THz. We use this model to calculate the dependency of the third harmonic response of BLG on the graphene temperature and the central frequency of the pulse. The current densities and the generated harmonics are calculated numerically for two values of the temperature, 10 and 100 K. We find that the ratio of the amplitude of the third harmonic to the fundamental in the reflected field is reduced by an order of magnitude as the temperature is increased from 10 to 100 K. Finally, we examine the nonlinear response at a number of central THz frequencies. We find that as the central frequency increases from 1 to 5 THz, there is an increase in the ratio of the third harmonic to the fundamental; reaching a maximum at a central frequency of 2 THz.

The paper is organized as follows. In section \ref{sec:2}, we present the derivation of our theoretical model. A tight-binding method is used to obtain dipole matrix elements within the length gauge. The matrix elements are then used in a four-band density matrix formalism in order to calculate the interband and intraband current densities. In section \ref{sec:3}, we present both the linear and  nonlinear THz response of numerical simulations for undoped BLG at different temperatures and central THz frequencies. The conclusions are presented in section \ref{sec:4}.

\section{Theory}\label{sec:2}
The calculations that we perform are based on a theoretical approach employing a density-matrix formalism in the length gauge.  A nearest-neighbor tight-binding model is used to treat the $\pi$-electrons in the graphene, which are taken to provide the conduction electrons only \cite{sarma2011electronic}.
\subsection{Energy Bands}
The tight-binding expression for the Bloch states is given by
\begin{equation}\begin{aligned}\psi_{n\mathbf{k}}(\mathbf{r}) & =A_{n}\left(\mathbf{k}\right)\sum_{i}\sum_{\mathbb{\mathbf{R}}}C_{i}^{n}\left(\mathbf{k}\right)\varphi_{pz}\left(\mathbf{r}-\mathbf{R-r}_{i}\right)e^{i\mathbf{k}\cdot\mathbf{R}},\end{aligned}
\end{equation}
where $A_{n}\left(\mathbf{k}\right)$ is a normalization factor, $n$ labels the conduction and valence bands, and the sum is over the Bravais lattice vectors $\mathbf{R}$. The sublattice coefficients, $C_{i}^{n}\left(\mathbf{k}\right)$,
 are associated with the four carbon atoms within the unit cell; the $\varphi_{pz}\left(\mathbf{r}\right)$ are the $2p_{z}$ orbitals of carbon. The index $i$ indicates a sum over the basis vectors $\mathbf{r}_{A_{1}},\,\mathbf{r}_{B_{1}},\,\mathbf{r}_{A_{2}},\,\mathbf{r}_{B_{2}}$, which give the position of sublattice sites $A_{1}$ and $B_{1}$  in the top layer, and $A_{2}$ and $B_{2}$ in the bottom layer (see Fig. \ref{fig:lattice2}). Explicitly, they are given by $\mathbf{r}_{A_{1}}=0$, $\mathbf{r}_{B_{1}}=a_{o}\widehat{\mathbf{x}}$, $\mathbf{r}_{A_{2}}=-a_{o}\widehat{\mathbf{x}}$ and $\mathbf{r}_{B_{2}}=0$.
\begin{figure}[h]
\centering
\includegraphics[width=0.35\textwidth]{lattice2.eps}
\caption{Schematic diagram of the crystal structure of BLG and the labelling conventions for atoms in the unit cell. A sites are in white, while B sites are in black\label{fig:lattice2}.}
\end{figure}

 The vector of the sublattice coefficients is given by $$\Bigl|\mathbf{k}_{n}\Bigr\rangle^{\dagger}=\left(C_{A_{1}}^{n*}\left(\mathbf{k}\right),\, C_{B_{1}}^{n*}\left(\mathbf{k}\right),\, C_{A_{2}}^{n*}\left(\mathbf{k}\right),\, C_{B_{2}}^{n*}\left(\mathbf{k}\right)\right).$$ Within the basis of these vectors, the secular equation for BLG can be expressed as
\begin{equation}
\textrm{det}\left[\begin{array}{cccc}
-\epsilon_{n\mathbf{k}} & f\left(\mathbf{k}\right)t_{\parallel} & 0 & t_{\perp}\\
f\left(\mathbf{k}\right)^{*}t_{\parallel} & -\epsilon_{n\mathbf{k}} & 0 & 0\\
0 & 0 & -\epsilon_{n\mathbf{k}} & f\left(\mathbf{k}\right)t_{\parallel}\\
t_{\perp} & 0 & f\left(\mathbf{k}\right)^{*}t_{\parallel} & -\epsilon_{n\mathbf{k}}
\end{array}\right]=0,\label{eq:sec}
\end{equation}
\vspace{3mm}
with the eigenvalue of band $n$ given by $\epsilon_{n\mathbf{k}}$.

The \textit{intralayer} hopping energy, $t_{\parallel}$, and the \textit{interlayer} hopping energy, $t_{\perp}$, are approximately equal to 3.03 eV and 0.3 eV, respectively \cite{malard2007probing}. The function $f\left(\mathbf{k}\right)\equiv(1+e^{-i\mathbf{k}\cdot\mathbf{a}_{1}}+e^{-i\mathbf{k}\cdot\mathbf{a}_{2}})$, is a result of the nearest neighbor intralayer electron hopping, where the $\mathbf{a}_i$ are the primitive translation vectors of graphene, given explicitly by \begin{equation}\begin{aligned}\mathbf{a}_{1} & =\frac{3a_{o}}{2}\widehat{\mathbf{x}}+\frac{\sqrt{3}a_{o}}{2}\widehat{\mathbf{y}},\end{aligned}
 \begin{aligned}\mathbf{a}_{2} & =\frac{3a_{o}}{2}\widehat{\mathbf{x}}-\frac{\sqrt{3}a_{o}}{2}\widehat{\mathbf{y}}.\end{aligned}\end{equation} Here $a_{o}$ is the nearest-neighbour separation $(a_{o}\simeq1.42\, \angstrom)$, not the length of the primitive vectors.

Equation (\ref{eq:sec}) can be solved exactly to yield expressions for the eigenvalues of BLG. In order of increasing energy, these are given by
\begin{equation}\begin{aligned}E_{v_{1}}=-\frac{\tilde{\epsilon}\left(\mathbf{k}\right)+t_{\perp}}{2} &  & E_{v_{2}}=-\frac{\tilde{\epsilon}\left(\mathbf{k}\right)-t_{\perp}}{2}\\
\\
E_{c_{1}}=\frac{\tilde{\epsilon}\left(\mathbf{k}\right)-t_{\perp}}{2} &  & E_{c_{2}}=\frac{\tilde{\epsilon}\left(\mathbf{k}\right)+t_{\perp}}{2},
\end{aligned}\label{eq:eig}\end{equation} where the first two expressions represent the valence bands, the second two the conduction bands and where we define $\tilde{\epsilon}\left(\mathbf{k}\right)=\sqrt{t_{\perp}^{2}+4|f\left(\mathbf{k}\right)|^{2}.}$

Due to the symmetry between the sublattices, the conduction and valence states in graphene are degenerate at two \textit{Dirac points}, given by: \begin{equation}\begin{aligned}\mathbf{K}a_{o} & =\frac{4\pi}{3\sqrt{3}}\widehat{\mathbf{y}},\\
\mathbf{K}^{\prime}a_{o} & =\frac{8\pi}{3\sqrt{3}}\widehat{\mathbf{y}}.
\end{aligned}\end{equation}
For energies within a few hundred meV of the Dirac points, we can expand the crystal momentum around the Dirac points as $\mathbf{k}=\mathbf{K}+\delta\mathbf{k}$ and $\mathbf{k}^{\prime}=\mathbf{K}^{\prime}+\delta\mathbf{k}$, where $\delta\mathbf{k}=k_{x}\hat{\mathbf{x}}+k_{y}\hat{\mathbf{y}}$. With this expansion, we can express the band energy near both Dirac points as $\tilde{\epsilon}\left(\mathbf{k}\right)$ as \begin{equation}\tilde{\epsilon}\left(\mathbf{K}+\delta\mathbf{k}\right)\approx\sqrt{t_{\perp}^{2}+\alpha k^{2}},\end{equation}where $k$ is the magnitude of the crystal momentum $k\equiv|\delta\mathbf{k}|$, $\alpha=4\hbar^{2}v_{F}^{2}$, where $v_{F}=3a_{o}t_{\perp}/2\hbar$ is the Fermi velocity. The energy dispersion given in Eq. (\ref{eq:eig}) for the four bands is shown in Fig. \ref{fig:1} for $k$-vectors near the $\mathbf{K}$-Dirac point.

\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{paper3.eps}
\caption{Band structure of unbiased bilayer graphene. Lower energy bands touch at the Dirac point; higher energy bands are separated by $2t_{\perp}$\label{fig:1}.}
\end{figure}

Using the expressions for the eigenvalues in Eq. (\ref{eq:eig}), we may also solve for the corresponding eigenstates exactly, to obtain

\begin{equation}
\Bigl|\mathbf{k}_{v_{1}}\Bigr\rangle=\frac{1}{2}\left(\begin{array}{c}
-\left(\frac{\tilde{\epsilon}+t_{\perp}}{\tilde{\epsilon}}\right)^{1/2}\\
\left(\frac{\tilde{\epsilon}-t_{\perp}}{\tilde{\epsilon}}\right)^{1/2}e^{-i\chi}\\
-\left(\frac{\tilde{\epsilon}-t_{\perp}}{\tilde{\epsilon}}\right)^{1/2}e^{i\chi}\\
\left(\frac{\tilde{\epsilon}+t_{\perp}}{\tilde{\epsilon}}\right)^{1/2}
\end{array}\right),
\end{equation}

\begin{equation}
\Bigl|\mathbf{k}_{v_{2}}\Bigr\rangle=\frac{1}{2}\left(\begin{array}{c}
\left(\frac{\tilde{\epsilon}-t_{\perp}}{\tilde{\epsilon}}\right)^{1/2}\\
-\left(\frac{\tilde{\epsilon}+t_{\perp}}{\tilde{\epsilon}}\right)^{1/2}e^{-i\chi}\\
-\left(\frac{\tilde{\epsilon}+t_{\perp}}{\tilde{\epsilon}}\right)^{1/2}e^{i\chi}\\
\left(\frac{\tilde{\epsilon}-t_{\perp}}{\tilde{\epsilon}}\right)^{1/2}
\end{array}\right),
\end{equation}

\begin{equation}
\Bigl|\mathbf{k}_{c_{1}}\Bigr\rangle=\frac{1}{2}\left(\begin{array}{c}
-\left(\frac{\tilde{\epsilon}-t_{\perp}}{\tilde{\epsilon}}\right)^{1/2}\\
-\left(\frac{\tilde{\epsilon}+t_{\perp}}{\tilde{\epsilon}}\right)^{1/2}e^{-i\chi}\\
\left(\frac{\tilde{\epsilon}+t_{\perp}}{\tilde{\epsilon}}\right)^{1/2}e^{i\chi}\\
\left(\frac{\tilde{\epsilon}-t_{\perp}}{\tilde{\epsilon}}\right)^{1/2}
\end{array}\right),
\end{equation}

\begin{equation}
\Bigl|\mathbf{k}_{c_{2}}\Bigr\rangle=\frac{1}{2}\left(\begin{array}{c}
\left(\frac{\tilde{\epsilon}+t_{\perp}}{\tilde{\epsilon}}\right)^{1/2}\\
\left(\frac{\tilde{\epsilon}-t_{\perp}}{\tilde{\epsilon}}\right)^{1/2}e^{-i\chi}\\
\left(\frac{\tilde{\epsilon}-t_{\perp}}{\tilde{\epsilon}}\right)^{1/2}e^{i\chi}\\
\left(\frac{\tilde{\epsilon}+t_{\perp}}{\tilde{\epsilon}}\right)^{1/2}
\end{array}\right),
\end{equation}
where $ e^{i\chi\left(\mathbf{k}\right)}=f\left(\mathbf{k}\right)/\left\vert f\left(\mathbf{k}\right)\right\vert$, and we have supressed the explicit $\mathbf{k}$-dependencies for simplicity.

As discussed in previous studies \cite{al2014high, virk2007semiconductor, aversa1995nonlinear}, modelling the nonlinear response of semiconductors in a limited band model is only expected to be accurate if one uses the length gauge. If instead the velocity gauge is employed, unphysical low-frequency divergences arise in the nonlinear response. As these divergences affect the response in the THz regime, one must develop sum rules in order to remove them. Moreover, trying to obtain the response to higher and higher order in the field, these sum rules become analytically intractable. Consequently, when using a limited basis of bands, the length gauge is advantageous. Hence, to model the THz interaction of BLG, we employ the length gauge Hamiltonian, expressed as $H=H_{0}-e\mathbf{r}\cdot\mathbf{E}$, where $H_{0}$ is the full Hamiltonian of
unperturbed BLG, $e = -|e|$ is the charge of an electron, $\mathbf{r}$ is the electron position vector, and $\mathbf{E}(t)$ is the THz electric field at the graphene. Given that we only consider normally incident plane waves, the field is uniform over the graphene sheets.

The carrier dynamics in BLG are calculated by solving the equations of motion for the density matrix in the basis of the Bloch states describing the four bands of BLG. For these, we require the matrix elements of the Hamiltonian between the various Bloch states:

\begin{equation}\Bigl\langle\mathbf{k}_{n}\Bigr|H\Bigl|\mathbf{k}_{m}^{\prime}\Bigr\rangle=E_{n}(\mathbf{k})\delta(\mathbf{k}-\mathbf{k}^{\prime})\mathbf{\delta}_{nm}-e\Bigl\langle\mathbf{k}_{n}\Bigr|\mathbf{r}\Bigl|\mathbf{k}_{m}^{\prime}\Bigr\rangle\cdot\mathbf{E}(t).\label{eq:9}
 \end{equation}

The matrix elements of $\mathbf{r}$ between Bloch states can be shown to be given by \cite{al2014high, aversa1995nonlinear}

\begin{equation}\left\langle n,\mathbf{k}\right.\left|\mathbf{r}\right.\left|m,\mathbf{k}^{\prime}\right\rangle =\delta(\mathbf{k}-\mathbf{k}^{\prime})\mathbf{\xi}_{nm}(\mathbf{k})+i\delta_{nm}\mathbf{\nabla_{k}}\delta(\mathbf{k}-\mathbf{k}^{\prime}),
 \label{eq:1}\end{equation} where the connection elements are defined as

\begin{equation}\mathbf{\xi}_{nm}(\mathbf{k})=\frac{\left(2\pi\right)^{2}i}{\Omega}\int_{\Omega}d^{3}\mathbf{r}u_{n,\mathbf{k}}^{\ast}\left(\mathbf{r}\right)\mathbf{\nabla}_{\mathbf{k}}u_{m,\mathbf{k}}\left(\mathbf{r}\right),
\end{equation} where $\Omega$ is the volume of a unit cell and  $u_{n\mathbf{k}}(\mathbf{r})$ is the periodic part of the Bloch function. These connection elements have been calculated using our tight-binding wave function. Ignoring the overlap of atomic wave functions on different atoms, near the $\mathbf{K}$-Dirac point the interband connection elements can be shown to be given by

\begin{equation}
\begin{aligned}\xi_{c_{1}v_{2}}\left(\mathbf{K}+\delta\mathbf{k}\right) & =\left(\frac{\tilde{\epsilon}+t_{\perp}}{2\tilde{\epsilon}}\right)\frac{\widehat{\mathbf{\theta}}}{k}\\
\xi_{c_{2}v_{1}}\left(\mathbf{K}+\delta\mathbf{k}\right) & =\left(\frac{\tilde{\epsilon}-t_{\perp}}{2\tilde{\epsilon}}\right)\frac{\widehat{\mathbf{\theta}}}{k}\\
\xi_{c_{1}v_{1}}\left(\mathbf{K}+\delta\mathbf{k}\right) & =-\frac{i\sqrt{\alpha}t_{\perp}}{2\tilde{\epsilon}^{2}}\hat{\mathbf{k}}\\
\xi_{c_{2}c_{1}}\left(\mathbf{K}+\delta\mathbf{k}\right) & =-\frac{\sqrt{\alpha}}{2\tilde{\epsilon}}\widehat{\mathbf{\mathbf{\theta}}}\\
\xi_{c_{2}v_{2}}\left(\mathbf{k}\right) & =\xi_{c_{1}v_{1}}^{*}\left(\mathbf{k}\right)\\
\xi_{v_{2}v_{1}}\left(\mathbf{k}\right) & =\xi_{c_{2}c_{1}}\left(\mathbf{k}\right),
\end{aligned}\label{eq:zeta}\end{equation} where $\delta\hat{\mathbf{k}}=cos(\theta)\hat{\mathbf{x}}+sin(\theta)\hat{\mathbf{y}}$ and $\hat{\theta}=-sin(\theta)\hat{\mathbf{x}}+cos(\theta)\hat{\mathbf{y}}$ are, respectively, the radial and angular unit vectors in cylindrical coordinates with the origin at the $\mathbf{K}$-Dirac point. Around the $\mathbf{K}^{\prime}$-Dirac point, we find that all the connection elements change sign except for those with components in the $\hat{\mathbf{k}}$-direction. Explicitly, we have
\begin{equation}
\begin{aligned}
\xi_{c_{1}v_{2}}\left(\mathbf{K}^{\prime}+\delta\mathbf{k}\right) & =-\xi_{c_{1}v_{2}}\left(\mathbf{K}+\delta\mathbf{k}\right)\\
\xi_{c_{2}v_{1}}\left(\mathbf{K}^{\prime}+\delta\mathbf{k}\right) & =-\xi_{c_{2}v_{1}}\left(\mathbf{K}+\delta\mathbf{k}\right)\\
\xi_{c_{2}c_{1}}\left(\mathbf{K}^{\prime}+\delta\mathbf{k}\right) & =-\xi_{c_{2}c_{1}}\left(\mathbf{K}+\delta\mathbf{k}\right)\\
\xi_{c_{1}v_{1}}\left(\mathbf{K}^{\prime}+\delta\mathbf{k}\right) & =\xi_{c_{1}v_{1}}\left(\mathbf{K}+\delta\mathbf{k}\right).
\end{aligned}\label{eq:zeta2}
\end{equation}

 Similarly, we can show that all the intraband connection elements - which are identical to the Berry connections of the respective bands \cite{chang1996berry, xiao2010berry} -  are zero, i.e.
\begin{equation}\xi_{nn}\left(\mathbf{k}\right)\equiv\mathcal{A}_{n}\left(\mathbf{k}\right)=0.\end{equation} This is in agreement with the idea that the Berry phase of BLG does not have to be simply $2\pi$, but rather can be an integer multiple of $2\pi$ \cite{park2011berry}. The vanishing Berry connection leads directly to a zero Berry curvature for each of the four bands; which is expected due to inversion symmetry present in BLG.

The dynamic equations for the density matrix elements can be found using the approach of Aversa and Sipe \cite{aversa1995nonlinear}, which was used in similar work on MLG \cite{al2014high}. These equations are given by
\begin{equation}
\begin{alignedat}{1}\frac{d\rho_{nm}(\mathbf{k})}{dt} & =i\frac{e}{\hbar}\mathbf{E}(t)\cdot\sum_{l}(\mathbf{\xi}_{nl}(\mathbf{k})\rho_{lm}(\mathbf{k})-\rho_{nl}(\mathbf{k})\mathbf{\xi}_{lm}(\mathbf{k}))\\
 & -\frac{e}{\hbar}\mathbf{E}(t)\cdot\nabla_{\mathbf{k}}\rho_{nm}(\mathbf{k})-i\omega_{nm}(\mathbf{k})\rho_{nm}(\mathbf{k})\\
 & -\frac{\rho_{nm}\left(\mathbf{k}\right)-\delta_{nm}\rho_{nm}^{eq}(\mathbf{k})}{\tau_{nm}},
\end{alignedat}
 \end{equation} where $n,m=\{c_{1},c_{2},v_{1},v_{2}\}$, $ \omega_{nm}(\mathbf{k})=\omega_{n}(\mathbf{k})-\omega_{m}(\mathbf{k})=(E_{n}(\mathbf{k})-E_{m}(\mathbf{k}))/\hbar,$ and $\rho_{nm}^{eq}(\mathbf{k})=f_{n}(\mathbf{k},T)\delta_{nm}$ is the carrier population in equilibrium when $n=m$, and is zero otherwise; where $f_{n}(\mathbf{k},T)$ is the Fermi-Dirac distribution at temperature $T$. In our numerical work, we model the vacancy populations rather than the valence band electrons to allow us to only include states near the Dirac point, which greatly reduces computation time.

Previous studies have shown that the scattering times in graphene are on the order of tens of femtoseconds \cite{bowlan2014ultrafast, paul2013high}, therefore to accurately model the THz response we must take into account scattering processes. At the low carrier densities considered in this work, carrier-carrier scattering is expected to be relatively unimportant, and the dominant scattering processes will be defect scattering and electron-phonon scattering. To account for these mechanisms in our model, we treat scattering phenomenologically. We introduce an interband decoherence time, $\tau_{nm}$, for the interband coherences, $\rho_{nm}(\mathbf{k})$, where $n\neq m$. We assume the decoherence time to be independent of $\mathbf{k}$. The populations, $\rho_{nn}(\mathbf{k})$, we take to relax back to Fermi-Dirac thermal distributions, $f_{n}(\mathbf{k},T)$, with relaxation times, $\tau_{n}$. Since the interaction with THz radiation induces interband transitions, as the simulation proceeds, we adjust the temperature of the Fermi-Dirac distribution so that the carriers relax to the instantaneous carrier populations. Also, we neglect interband relaxation as it has been found that the intraband scattering times are much shorter than the time taken for conduction band electrons to relax to the valence band \cite{tielrooij2013}.

We employ a direct computational approach to solve the above equations, wherein we put $\mathbf{k}$ on a grid and step through time using a Runge-Kutta algorithm. To facilitate this, we make use of balanced difference approximations to the gradients.

Now that we have the dynamic equations for the density matrix elements, we turn to determining the expression for the current density. Following the formalism of Aversa and Sipe \cite{aversa1995nonlinear} the current density can be expressed as
\begin{equation}\mathbf{J}(t)=\frac{e}{m}\text{Tr}\left\{ \mathbf{p}\rho(t)\right\}.\end{equation} Using the fact that $\frac{\mathbf{p}}{m}=\frac{1}{i\hbar}\left[\mathbf{r},H\right],$ and decomposing the position operator into intraband and interband parts, $\mathbf{r}=\mathbf{r}_{i}+\mathbf{r}_{e},$ we can write this as \cite{aversa1995nonlinear}
\begin{equation}\begin{aligned}\mathbf{J}(t) & =\frac{e}{i\hbar}\text{Tr}\left\{ \left[\mathbf{r},H\right]\rho(t)\right\} \\
 & =\frac{e}{i\hbar}\sum_{n}\sum_{\mathbf{k}}\left\langle n,\mathbf{k}\right.|\left[\mathbf{r}_{i},H\right]\rho(t)\left|n,\mathbf{k}\right\rangle \\
 & +\frac{e}{i\hbar}\sum_{n}\sum_{\mathbf{k}}\left\langle n,\mathbf{k}\right.|\left[\mathbf{r}_{e},H\right]\rho(t)\left|n,\mathbf{k}\right\rangle,
\end{aligned}\end{equation} where the trace is over the single electron states, and $\rho(t)$ is the reduced density matrix with matrix elements $\rho_{nm}(\mathbf{k})$. The decompostion of the position operator allows us to define the total current density as the sum of an intraband contribution, $\mathbf{J}_{i}$, and an interband contribution, $\mathbf{J}_{e}$. Using our expression for the Hamiltonian, as well as the matrix elements of the position operator (Eq. \ref{eq:1}), one may determine expressions for these contributions. The procedure is similar to that presented in recent work on MLG \cite{al2015nonperturbative}. After considerable work, the intraband current density near the $\mathbf{K}$-Dirac point can be shown to be given by

\begin{equation}\begin{aligned}\mathbf{J_{i}} & =\frac{\alpha e}{2\hbar}\sum_{\mathbf{k}}\frac{\mathbf{k}}{\tilde{\epsilon}}\left\{ \rho_{c_{1}c_{1}}(\mathbf{k})+\rho_{c_{2}c_{2}}(\mathbf{k})+\rho_{h_{1}h_{1}}(\mathbf{\mathbf{k}})+\rho_{h_{2}h_{2}}(\mathbf{\mathbf{k}})\right\} \\
 & -\frac{2e^{2}}{\hbar}\sum_{\mathbf{k}}\textrm{Re}\left\{ \rho_{c_{1}v_{2}}(\mathbf{k})\mathbf{\nabla_{\mathbf{k}}}\left(\mathbf{E}(t)\cdot\mathbf{\xi_{v_{2}c_{1}}}(\mathbf{\mathbf{k}})\right)\right.\\
 & +\rho_{c_{2}v_{1}}(\mathbf{k})\mathbf{\nabla_{\mathbf{k}}}\left(\mathbf{E}(t)\cdot\mathbf{\xi_{v_{1}c_{2}}}(\mathbf{\mathbf{k}})\right)\\
 & +\left[\rho_{c_{2}v_{2}}(\mathbf{\mathbf{k}})-\rho_{c_{1}v_{1}}(\mathbf{\mathbf{k}})\right]\mathbf{\nabla_{\mathbf{k}}}\left(\mathbf{E}(t)\cdot\mathbf{\xi_{v_{2}c_{2}}}(\mathbf{k})\right)\\
 & +\left.\left[\rho_{c_{2}c_{1}}(\mathbf{\mathbf{k}})+\rho_{v_{2}v_{1}}(\mathbf{k})\right]\mathbf{\nabla_{\mathbf{k}}}\left(\mathbf{E}(t)\cdot\mathbf{\xi_{c_{1}c_{2}}}(\mathbf{k})\right)\right\},
\end{aligned}\label{eq:3}\end{equation} while the interband current density is given by

\begin{equation}\begin{aligned}\mathbf{J}_{e} & =2e\sum_{\mathbf{k}}\text{Re}\left\{ \left(\frac{\tilde{\epsilon}-t_{\perp}}{2\tilde{\epsilon}}\right)\frac{\left(-k_{y}\hat{\mathbf{x}}+k_{x}\hat{\mathbf{y}}\right)}{k^{2}}\dot{\rho}_{c_{1}v_{2}}(\mathbf{k})\right.\\
 & +\left(\frac{\tilde{\epsilon}+t_{\perp}}{2\tilde{\epsilon}}\right)\frac{\left(-k_{y}\hat{\mathbf{x}}+k_{x}\hat{\mathbf{y}}\right)}{k^{2}}\dot{\rho}_{c_{2}v_{1}}(\mathbf{k})\\
 & +\frac{i\sqrt{\alpha}t_{\perp}}{2\tilde{\epsilon}^{2}}\frac{k_{x}\hat{\mathbf{x}}+k_{y}\hat{\mathbf{y}}}{k}\left(\dot{\rho}_{c_{1}v_{1}}(\mathbf{k})-\dot{\rho}_{c_{2}v_{2}}(\mathbf{\mathbf{k}})\right)\\
 & -\left.\frac{\sqrt{\alpha}}{2\tilde{\epsilon}}\frac{\left(-k_{y}\hat{\mathbf{x}}+k_{x}\hat{\mathbf{y}}\right)}{k}\left(\dot{\rho}_{c_{2}c_{1}}(\mathbf{k})+\dot{\rho}_{v_{2}v_{1}}(\mathbf{\mathbf{k}})\right)\right\}.
\end{aligned}\label{eq:4}\end{equation} The sums over $\mathbf{k}$ in the current density expressions are restricted to a region near the $\mathbf{K}$-Dirac point, and we include a factor of 2 to account for spin degeneracy.
We also need to take into account the current density near the $\mathbf{K}^{\prime}$ point. Similar to MLG, we find that due to the symmetry of the Brillouin zone, the current densities around both Dirac points are identical. To obtain the total current density of BLG, we then only need to multiply the results calculated at the $\mathbf{K}$-point by two in order to account for this degeneracy.

In what follows, we consider a suspended BLG sample, and use the time-dependent current densities to calculate the transmitted and the reflected THz fields, using a procedure identical to that used for MLG \cite{al2015nonperturbative}. We show in the next section that for low amplitude fields, we obtain the expected linear conductivities, which relate the THz field at the graphene sheet to the induced current densities. Furthermore, we have verified convergence in the nonlinear regime by altering the grid density, the extent of the grid, the time-step tolerance, and the polarization of the incident field.

\subsection{Linear Response}
In order to calculate the linear response of BLG to an incident field, we need to calculate expressions for the density matrix elements to first order in the THz field. Once we have these, we may then use Eq. (\ref{eq:3}) to express the first order \textit{intraband} current density as
\begin{equation}\begin{aligned}\mathbf{J_{i}}^{(1)} & =\end{aligned}
C(\omega_{p})\int_{0}^{\infty}d\tilde{\epsilon}^{\prime}\left(1+\frac{t_{\perp}^{2}}{\left(\tilde{\epsilon}^{\prime}+t_{\perp}\right)^{2}}\right)\sum_{n}f_{FD}(E_{n}),\label{eq:intra}\end{equation}
where $\tilde{\epsilon}^{\prime}$ is a shifted energy, given by $\tilde{\epsilon}^{\prime}=\tilde{\epsilon}-t_{\perp}$. $C(\omega_{p})$ is a frequency-dependent coefficient that includes the electric field: $$C(\omega_{p})=\frac{i|e|^{2}\mathbf{E}\left(\omega_{p}\right)e^{-i\omega_{p}t}}{2\hbar^{2}\pi\left(\omega_{p}+i/\tau_{c}\right)},$$ where we take our field to be harmonic, $$\mathbf{E}\left(t\right)=\mathbf{E}\left(\omega_{p}\right)e^{-i\omega_{p}t}.$$  In Eq. (\ref{eq:intra}), the sum is over the Fermi-Dirac distributions of the electrons and holes in the four bands of BLG. This is given expilicity by
\begin{equation}\begin{aligned}\sum_{n}f_{FD}(E_{n}) & =\frac{1}{1+e^{\beta(\frac{\tilde{\epsilon}^{\prime}}{2}-\mu)}}+\frac{1}{1+e^{\beta(\frac{\tilde{\epsilon}^{\prime}}{2}+\mu)}}\\
 & +\frac{1}{1+e^{\beta(\frac{\tilde{\epsilon}^{\prime}}{2}+t_{\perp}-\mu)}}+\frac{1}{1+e^{\beta(\frac{\tilde{\epsilon}^{\prime}}{2}+t_{\perp}+\mu)}},
\end{aligned}\end{equation} where $\beta=k_{B}T$ and $\mu$ is the chemical potential of the system.

As a check, we may evaluate the first order intraband current density in the limit of zero interlayer coupling. In this limit, $t_{\perp}\rightarrow0$, only the integral of the sum over the distributions remains. Thus our intraband current density reduces to
\begin{equation}\begin{aligned}\begin{aligned}\mathbf{J_{i}}^{(1)} & =C(\omega_{p})4k_{B}T\left\{ \text{ln}\biggl(1+e^{\beta\mu}\biggr)+\text{ln}\biggl(1+e^{-\beta\mu}\biggr)\right\} \\
 & =C(\omega_{p})8k_{B}T\text{ln}\left[2\text{Cosh}(\beta\mu/2)\right].
\end{aligned}\end{aligned}\end{equation} From here it is easy to read off the first order intraband conductivity as being
\begin{equation}\begin{aligned}\lim_{t_{\perp} \to 0}\sigma_{i}^{(1)}(\omega) & =\frac{4i|e|^{2}k_{B}T}{\hbar^{2}\pi\left(\omega_{p}+i/\tau_{c}\right)}\text{ln}\Bigl[2\text{Cosh}(\beta\mu/2)\Bigr].\end{aligned}\end{equation}
This is the previously found result for the monolayer reponse \cite{mikhailovnonlin2009}, multiplied by two to account for the two layers of BLG.

For BLG ($t_{\perp} \neq 0$), the integral in Eq. (\ref{eq:intra}) must be evaluated numerically in general, however in the limit that $T\rightarrow0$, it may be evaluated analytically. In this limit the Fermi-Dirac distributions will behave as Heaviside step functions; having the effect that only energies lower than the Fermi level, $E_{F}$, contribute to the integral. Assuming that our Fermi level is non-zero and located in the conduction bands, we only get contributions from the $\rho_{c_{1}c_{1}}\left(\mathbf{k}\right)$ and $\rho_{c_{2}c_{2}}\left(\mathbf{k}\right)$ distributions. In this case, we obtain two distinct contributions to the intraband current density. Thus, we are able to determine the intraband conductivities of both conduction bands in the limit of zero temperature. For the low-energy band $c_{1}$, this is given by \begin{equation} \lim_{T \to 0}\sigma_{i}^{(1)}(\omega)_{c_{1}}=\frac{i|e|^{2}E_{F}}{\hbar^{2}\pi\left(\omega_{p}+i/\tau_{c}\right)}\frac{(E_{F}+t_{\perp})}{(E_{F}+\frac{t_{\perp}}{2})},\end{equation} while for the high-energy band $c_{2}$, we have \begin{equation}\lim_{T \to 0}\sigma_{i}^{(1)}(\omega)_{c_{2}}=\frac{i|e|^{2}E_{F}}{\hbar^{2}\pi\left(\omega_{p}+i/\tau_{c}\right)}\frac{(E_{F}-t_{\perp})}{(E_{F}-\frac{t_{\perp}}{2})}\theta(E_{F}-t_{\perp}).\end{equation} Here the step function assures that the Fermi level must be greater than $t_{\perp}$ for $\rho_{c_{2}c_{2}}\left(\mathbf{k}\right)$ to provide a contribution.

Similarly, we may now use Eq. (\ref{eq:4}), and the connection elements from Eq. (\ref{eq:zeta}), to express the first order \textit{interband} current density as
\begin{equation}\begin{aligned}\mathbf{J}_{e}^{(1)} & =D(\omega_{p})\int_{0}^{\infty}d\tilde{\epsilon}^{\prime}\left\{ \left[\frac{(\tilde{\epsilon}^{\prime}+2t_{\perp})}{\tilde{\epsilon}^{\prime}\left(\tilde{\epsilon}^{\prime}+t_{\perp}\right)\left(\frac{\tilde{\epsilon}^{\prime}}{\hbar}-\omega_{p}-i/\tau\right)}\right.\right.\\
 & +\frac{t_{\perp}^{2}}{\left(\tilde{\epsilon}^{\prime}+t_{\perp}\right)^{3}\left(\frac{\tilde{\epsilon}^{\prime}+t_{\perp}}{\hbar}-\omega_{p}-i/\tau\right)}\\
 & -\left.\frac{1}{\left(\tilde{\epsilon}^{\prime}+t_{\perp}\right)\left(\frac{t_{\perp}}{\hbar}-\omega_{p}-i/\tau\right)}\right]\times R(\frac{\tilde{\epsilon}^{\prime}}{2})\\
 & +\left[\frac{\tilde{\epsilon}^{\prime}}{\left(\tilde{\epsilon}^{\prime}+t_{\perp}\right)\left(\tilde{\epsilon}^{\prime}+2t_{\perp}\right)\left(\frac{\tilde{\epsilon}+2t_{\perp}}{\hbar}-\omega_{p}-i/\tau\right)}\right.\\
 & +\frac{t_{\perp}^{2}}{\left(\tilde{\epsilon}^{\prime}+t_{\perp}\right)^{3}\left(\frac{\tilde{\epsilon}^{\prime}+t_{\perp}}{\hbar}-\omega_{p}-i/\tau\right)}\\
 & +\left.\left.\frac{1}{\left(\tilde{\epsilon}^{\prime}+t_{\perp}\right)\left(\frac{t_{\perp}}{\hbar}-\omega_{p}-i/\tau\right)}\right]\times R(\frac{\tilde{\epsilon}^{\prime}}{2}+t_{\perp})\right\} +c.c,
\end{aligned}\label{eq:inter}\end{equation} where $D(\omega_{p})$ is a frequency dependent coefficient given by $$D(\omega_{p})=\frac{-i\omega_{p}|e|^{2}}{4\pi\hbar}\mathbf{E}\left(\omega_{p}\right)e^{-i\omega_{p}t},$$ and the function $R(x)$ is dependent on the temperature, Fermi level and energy of the system, which is given explcitly by $$R(x)=\frac{\sinh\beta(x)}{\cosh\beta\mu+\cosh\beta(x)}.$$

We may evaluate the above integral analytically in the limit of zero temperature and scattering, where $T \to 0$ and $\tau \to \infty$, respectively. In this limit, the complex factors in each term reduce to a Dirac delta function - when we neglect the contribution of the principal part of the integral. Taking only the positive frequency portion of the above expression, we obtain for the first order interband conductivity
\begin{equation}\begin{aligned}\lim_{\mathclap{\substack{T \to 0 \\ \tau \to \infty}}}\sigma_{e}^{(1)}(\omega_{p}) & =\frac{|e|^{2}}{4\hbar}\left\{ \frac{(\hbar\omega_{p}+2t_{\perp})}{\left(\hbar\omega_{p}+t_{\perp}\right)}+\frac{2t_{\perp}^{2}}{\hbar^{2}\omega_{p}^{2}}\theta\left(\hbar\omega_{p}-t_{\perp}\right)\right.\\
 & +\left.\frac{\left(\hbar\omega_{p}-2t_{\perp}\right)}{\left(\hbar\omega_{p}-t_{\perp}\right)}\theta\left(\hbar\omega_{p}-2t_{\perp}\right)\right\}.
\end{aligned}\end{equation} This expression agrees with those found in the literature \cite{abergel2007optical, nicol2008optical}. We see that BLG exhibits a minimum conductivity that is associated with interband transitions. For very large incident frequencies, this minimum conductivity is given by $\frac{|e|^{2}}{2\hbar}$, which is twice that found in monolayer graphene, as expected.

We may also use Eqs. (\ref{eq:intra}) and (\ref{eq:inter}) to determine the first order current densities at non-zero temperatures and scattering times, which is a closer approximation of experimental conditions. We present the results of this analysis in the next section.

\section{Simulation Results}\label{sec:3}
We have shown that our analytic expressions for the linear THz response agrees with the literature in certain limits. Going beyond the linear regime to determine the response at higher order in the THz field is our main interest. We want to be able to examine the dependency of the THz response on the temperature and incident frequency of BLG. To achieve this aim we employ a simulation, which offers the ability to control the desired parameters, and calculate the interband and intraband current densities at the desired field amplitudes. The field transmitted from the BLG is calculated as a function of the current densities and the incident field. The transmitted field is then spectrally analyzed to determine its frequency components; high harmonic generation in the spectral composition indicating nonlinear behaviour.

We begin with a check on the first order calculations by comparing the linear conductivity derived from the first order equations of the previous section to that calculated by the full simulation. We then proceed to examine the higher order response of BLG, determining at which incident field amplitudes we might expect to see the largest generation of the third harmonic.

\subsection{Linear Results}
As an initial check, we compare on a single plot the conductivity arising from the full current density (intraband and interband), as calculated by both the simulation and the closed form expressions (Eqs. (\ref{eq:intra}) and (\ref{eq:inter})). We plot these results in Figs. \ref{fig:4.2} and \ref{fig:4.2-1} for Fermi levels of 60 and 360 meV respectively, which correspond to energies of $0.2t_{\perp}$ and $1.2t_{\perp}$. These values also allow us to make comparisons to similar work found in the literature \cite{nicol2008optical}. In both instances, the scattering time is $80\, fs$, and the temperature is 50 K. As can be seen the agreement between the simulated and first order results is excellent.

\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{fig4-2a.eps}
\caption{Comparison of the full linear conductivity, calculated by computer simulation and numerical integration of the closed form expression. Fermi level is $60\, meV\, (0.2t_{\perp})$, scattering time is $80\, fs$, and temperature is 50 K. Conductivity is measured in units of the universal conductivity of BLG, $\sigma_{0}=e^{2}/2\hbar$\label{fig:4.2}.}
\end{figure}

\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{fig4-3a.eps}
\caption{Comparison of the full linear conductivity, calculated by computer simulation and numerical integration of the closed form expression. Fermi level is $360\, meV\, (1.2t_{\perp})$, scattering time is $80\, fs$, and temperature is 50 K. Conductivity is measured in units of the universal conductivity of BLG, $\sigma_{0}=e^{2}/2\hbar$\label{fig:4.2-1}.}
\end{figure}

In both plots we see features at incident freqeuncies of $f=0$ and $f\simeq75$ THz. The zero frequency peak is a result of the intraband Drude response. This feature vanishes for zero Fermi level and temperature, as the intraband transitions become negligible in this limit. The second peak at $f\simeq 75$ THz is due to the difference in energy between the two conduction bands of BLG ($\simeq300\, meV$). Specifically, it arises from the perfect nesting between the dispersions of the higher and lower conduction bands, $c_{2}\,\&\, c_{1}$. These two bands sit directly on top of one another and differ in energy only by the constant displacement $t_{\perp}$. Thus, the nesting of the bands leads to a large spectral weight at frequencies close to $t_{\perp}$, which is approximately equivalent to 75 THz. The additional features present in the plots reflect other possible interband transitions, such as $v_{1}\rightarrow c_{1}$ or $v_{1}\rightarrow c_{2}$. These transitions require incident radiation of energy $t_{\perp}$and $2t_{\perp}$, respectively, and their spectral weights are largely dependent on the Fermi level.

\subsection{Nonlinear Results}
We now present our simulation results for the nonlinear THz response of BLG. In each of our simulations for BLG we took the sample to be undoped $(\mu=0)$. The response was then calculated at field amplitudes ranging from 0.5 kV/cm to 2.0 kV/cm at two different temperatures: 10 K and 100 K.  This is followed by the calculation of the response at several incident frequencies at a temperature of 10 K. For the system and fields modelled in this section, only the $v_{2}$ and $c_{1}$ bands contribute to the THz response.

Our input THz field is a sinusoidal Gaussian pulse with central frequency of 1 THz and full width at half maximum (FWHM) of 1 ps. Mathematically, this may be expressed as
\begin{equation}
\mathbf{E}_{i}(t)=\mathbf{E}_{0}e^{-4\textrm{log}(2)\left(\frac{t-t_{0}}{T_{FWHM}}\right)^{2}}\textrm{sin}\left[2\pi f_{0}(t-t_{0})\right],
 \end{equation}
where $t_{0}$ and $T_{FWHM}$ are the temporal shift and full width at half maximum of the Gaussian pulse, respectively. The central frequency of the pulse is given by $f_{0}$. We take our scattering and relaxation times to be fixed at 50 fs, which is an average of theoretical and measured values for such (sample-dependent) constants. This is a conservative estimate for undoped BLG at low temperature; an increase in this time will lead to stronger harmonic generation \cite{al2014high}.  In the simulation, we keep the Fermi level at the Dirac point and as carriers are injected we raise the temperature of the distribution to which the carriers relax to account for the increase in the carrier density.

\subsubsection*{T=10 K Results}
At a temperature of 10 K, the intrinsic thermal carrier density in BLG is approximately $1.4\times10^{10}/cm^{2}$ and the electrons have an average energy of 1.03 meV. In comparison, at this temperature MLG has an intrinsic carrier density of approximately $2.0\times10^{8}/cm^{2}$, and an average electron energy of 1.89 meV. Thus, we find a much larger thermal carrier density in BLG than we do in MLG. However, the average energy of the carriers is slightly larger in MLG. In both cases, the average electron energy is much less than the central photon energy of 4.14 meV associated with a 1 THz pulse, thus to first order in the field, interband transitions are largely unaffected by the thermal energy.

All of the carriers (injected and thermal) are driven within the conduction and valence bands by the applied electric field. The subsequent motion leads to a blocking and unblocking of the electronic states that are available for extra carrier injection. This results in an interplay between intraband and interband dynamics that is paramount in producing the nonlinear response \cite{al2014high}.

We now present results for a number of different field amplitudes.
In Fig. \ref{fig:4.3}, we plot the intraband and interband current densities for four different incident field amplitudes (0.5, 1, 1.5 and 2 kV/cm). All current densities are normalized to the peak value, $E_{0},$ of the incident field such that, if the response were linear, these relative currents would be unchanged by an increase in incident field. This allows for a comparison of the currents at each field amplitude, and for the clear identification of any nonlinear behaviour. In what follows, we refer to these as \textit{relative current densities}. The calculated relative intraband and interband current densities at these specific field amplitudes are shown in Figs. \ref{fig:4.3}(a) and \ref{fig:4.3}(b), respectively.
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{paper1.eps}
\caption{Response of BLG to a incident field of 1 THz at T=10K and for four different field amplitudes. a) Intraband current density normalized to incident field amplitude. b) Interband current density normalized to incident field amplitude.}\label{fig:4.3}
\end{figure}

Looking first at the intraband current density, we find that as the field amplitude is increased, the relative intraband current decreases. This is due to the `clipping` phenomena that is a property of linear dispersions, such as that found in MLG. Although BLG has a parabolic dispersion near the Dirac point, as we move away from this point, the dispersion becomes linear (see Fig. \ref{fig:1}). Thus, the high field amplitudes drive carriers far enough away from the Dirac point such that they occupy the linear part of the band structure. In this region, the carrier velocities become constant and the field is not able to drive them to higher velocities; the intraband current is clipped as a result.

We note, however, that the current clipping nonlinearity in the intraband current density is much smaller than the nonlinearity seen in the interband current. We also note that due to the large intrinsic carrier density of BLG due to the large number of states, carrier injection has only a minor effect on the intraband current density at these field amplitudes. This is in contrast to MLG, where one finds a substantial increase in the relative intraband current density arising from the injection of carriers \cite{al2014high}.

Examining next the interband current density, we observe large distortions for all of the field amplitudes. Based on the strength of these distortions, we expect the interband current to be a large source of nonlinearity at these field amplitudes. The motion of the charge carriers in the conduction band gives rise to an interplay between intraband and interband dynamics. We can visualize this interplay between interband and intraband motion via a plot of the electron density in the conduction band, as shown in Fig. \ref{fig:density}.
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{paperdensity.eps}
\caption{Electron density distribution in k-space (a) for the initial thermal distribution and (b) at time t=1.75 ps for a 0.5 kV/cm incident field. White lines indicate the position of the Dirac point.}\label{fig:density}
\end{figure}
As the pulse hits the BLG sample, the population of carriers in the conduction band is driven away from the Dirac point by the incident THz field; this driving of the carriers within each band is the source of the intraband current. Fig. \ref{fig:density}(a) shows the initial electron density before the pulse hits. In Fig. \ref{fig:density}(b) we see the density at t = 1.75 ps, where the magnitude of the amplitude of the incident field is a maximum. We can see the distribution has been driven to the right of the Dirac point. This intraband motion also opens up electron states near the Dirac point that were previously occupied before the pulse arrived. As a result of the reduction in Pauli blocking, carriers from the valence band $v_{2}$ may be injected into the conduction band $c_{1}$ near the Dirac point. This is evidenced in the dark, high-density regions that appear above and below the $k_{y}=0$ line to the right of the Dirac point in Fig. \ref{fig:density}(b). This carrier injection is the source of the interband current.

Thus, we see that there is an interesting relationship between the intraband and interband dynamics; the motion of carriers within each band has an effect on both interband and intraband current densities. Importantly, this relationship manifests itself in the nonlinear response of BLG.

We now plot the time-dependent reflected fields normalized to the peak value of the different incident field amplitudes in Fig. \ref{fig:4.3-1}(a), and the spectral responses normalized to the peak amplitude at the fundamental frequency (1 THz) in Fig. \ref{fig:4.3-1}(b).
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{paper2.eps}
\caption{Response of BLG to a incident field of 1 THz at T=10K for four different field amplitudes. a) The reflected field in the temporal domain, normalized to the amplitude of the incident field. Value of $E_r/E_0$ is multiplied by 100 for clarity. b) The amplitude spectra of the reflected signal normalized to the peak at the fundamental frequency of 1 THz.}\label{fig:4.3-1}
\end{figure}

In the upper plot we can see strong distortion of the reflected fields for all of the selected field amplitudes. Note that because there is no substrate the induced current is the entire source of the reflected field, which only has a peak amplitude of approximately 2 V/cm for the incident field of 1 kV/cm. To see harmonic content in the reflected field, Fig. \ref{fig:4.3-1}(b) we plot the Fourier transform of the reflected fields normalized to the peak reflected field at the fundamental frequency. We find that at the lowest field amplitude of 0.5 kV/cm, there is a clear third harmonic signal. Furthermore, we see a maximum in the third harmonic of the relfected field at an incident field amplitude of about 1.5 kV/cm. At this amplitude the third harmonic reaches a maximum of approximately 30\% of the reflected spectral peak at the fundamental. As we increase the incident field amplitude further, we begin to see a decrease in the third harmonic.

Moreover, at these high field amplitudes we also see the presence of the 5th harmonic, which is largely absent at lower field amplitudes. Its amplitude takes a maximum value of approximately 15\% the spectral peak at the fundamental for an incident field of 2 kV/cm. The appearance of the fifth harmonic indicates that at the higher field amplitudes fifth order processes are significant.  Just as was found with MLG \cite{al2014high}, the fifth order response results in a decrease in the third harmonic.

\subsubsection*{T=100 K Results}
We next present results for simulations at a temperature of 100 K. At this temperature, the intrinsic thermal carrier density is $3.4\times10^{10}/cm^{2}$ and the average thermal electron energy is 10.8 meV. This is compared to the average photon energy of 4.14 meV associated with a 1 THz pulse. Thus, the Pauli blocking of the interband transitions is increased due to the increased thermal populations of carriers; reducing the overall interband current density. Additionally, the increase in thermal carriers gives rise to an increase in the intraband current density. Thus, at this temperature the intraband current is dominant, resulting in a reduction in the interplay between intraband and interband dynamics, and ultimately a reduction in high harmonic generation.

We can see this by looking again at both the reflected field and the spectral response for each of the incident field amplitudes. The normalized time-dependent reflected fields for the different field amplitudes are shown in Fig. \ref{fig:4.4}(a), and the spectral responses normalized to the peak amplitude at the fundamental frequency (1 THz) are presented in Fig. \ref{fig:4.4}(b).
\begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth]{paper2T_100.eps}
\caption{Response of BLG to a incident field of 1 THz at T=100K for four different field amplitudes. a) The reflected field in the temporal domain, normalized to the amplitude of the incident field. Value of $E_r/E_0$ is multiplied by 100 for clarity. b) The amplitude spectra of the reflected signal normalized to the peak at the fundamental frequency of 1 THz.}\label{fig:4.4}
\end{figure}

In the upper plot we see far less distortion in the reflected fields than we do at T=10 K, for all of the selected field amplitudes. The reduction in distortion is again due to the increase in the thermal carrier density and subsequent increase in Pauli blocking. In turn this results in diminished harmonic generation,  which is shown in Fig. \ref{fig:4.4}(b). We find a maximum in the third harmonic of the relfected field at an incident field amplitude that is now approximately 1 kV/cm. At this amplitude it reaches a maximum of only approximately 2.8\% of the reflected peak at the fundamental, which is still greater than that found in MLG at the same temperature \cite{al2015nonperturbative}. At higher field amplitudes, we also see the presence of the 5th harmonic.  Its amplitude takes a maximum value of approximately 1.5\% of the peak at the fundamental for an incident field of 2 kV/cm.

Thus, we find that as the temperature of the system is increased, the nonlinear response - specifically, high harmonic generation - is greatly diminished as a result of the reduction in the interplay between interband and intraband dynamics. Therefore, if one is to observe high harmonic generation (HHG) experimentally in BLG, it is apparent that low temperatures are necessary. This allows for the maximization of the dynamic interplay that is paramount to the nonlinear repsonse.

\subsubsection*{Effect of Central Frequency}
Finally, we study the effect that the central frequency of the incident pulse has on the generation of the third harmonic. The interplay between intraband and interband dynamics has a strong dependence on the central frequency, because when the central frequency is increased, the carriers are injected farther from the Dirac point. In Fig.  \ref{fig:4.5} , we plot the normalized third harmonic amplitude as a function of the incident THz field amplitude for central frequencies of 1.0, 2.0 and 5.0 THz. In each case, we adjust the duration of the pulse such that the product of the central frequency and the pulse duration is constant, so that the pulse remains single-cycle for each of the central frequencies considered.

As the central frequency of the incident pulse is increased, we observe a maximum third harmonic amplitude that is larger in comparison to the simulations performed at 1 THz. For a central frequency of 2 THz, the normalized third harmonic amplitude peaks at approximately 53\% of the fundamental for a field amplitude of 2.5 kV/cm. This is compared to a value of 30\% for the central frequency of 1 THz at a field amplitude of 1.5 kV/cm.

As the pulse frequency is increased further to 5 THz, the maximum normalized third harmonic is found to decrease to 40\% of the fundamental. Furthermore, this maximum value is not obtained until the input field amplitude reaches 5 kV/cm.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{paperEvsA.eps}
\caption{Third harmonic amplitude spectra normalized to the peak of the fundamental central frequency, for three different incident frequencies: 1,\,2 and 5 THz.}\label{fig:4.5}
\end{figure}

The observed increase in the nonlinear interplay with increasing central frequency is a result of a number of factors. First, due to the approximately inverse relationship between the linear conductivity and the frequency, the intraband current density decreases as we increase the central frequency of the pulse. This in turn increases the ratio of the third harmonic to the fundamental, due to the reduction in the part of the reflected fundamental that arises due to the intraband current.

Additionally, we find that as the incident frequency is increased there is an increase in the interband current density. The increase arises because the interband transitions that are resonant with the 2 THz and 5 Thz photons are mostly unoccupied when the pulse arrives, whereas at lower frequencies they may be occupied by either thermal or injected carriers. This larger interband current density results in an increased  dynamic interplay, and thereby yields a larger third harmonic field.

\vspace{5mm}
\section{Summary}\label{sec:4}
We have presented a detailed study of the nonlinear response of unbiased bilayer graphene at THz frequencies. A theoretical model has been developed, which is based on the dynamic equations of density matrix elements, employing the length gauge. The model enabled the calculation of eigenvalues and eigenvectors of BLG, as well as interband and intraband connection elements. Expressions for interband and intraband current densities were also derived.

Through the use of simulation, we determined solutions of the density matrix dynamic equations. These solutions were then applied to the study of high harmonic generation in BLG. Investigating the effect of system parameters - such as the central THz frequency and the ambient temperature - on third harmonic generation was the main focus.

Our results show that for a temperature of 10 K, scattering time of 50\,fs and incident field of 1 THz, the third harmonic can be as large as 30\% of the fundamental for an incident THz field of 1.5 \textrm{kV/cm}. As we increase the temperature to 100 K, we find that the maximum third harmonic generation is reduced by an order of magnitude, due to the reduction in interband transitions.

Finally, we found that as the central THz frequency is increased from 1 to 5 THz, we see an increase in the third harmonic amplitude; reaching a maximum of 53\% of the fundamental for a 2 THz incident field at 2.5 kV/cm. These results may be of use to experimentalists aiming to probe the nonlinear response of bilayer graphene.

To experimentally observe the high harmonics we predict for BLG, one must consider the dynamic range of the THz spectrometers - defined as the ratio of the frequency dependent signal strength to the detected noise floor \cite{jepsen2005dynamic}. For a 1 THz  incident field of 1 kV/cm, we find the peak amplitude of the reflected field to be approximately 2.2 V/cm (53 dB less than the incident field). Thus, a detection technique that allows for a dynamic range larger than 53 dB is required for the measurement of the reflected signal. Such a dynamic range is experimentally feasible; a dynamic range of 90 dB has been reported in recent work \cite{vieweg2014terahertz}.

\bibliographystyle{apsrev4-1}

\section{Using \protectREV\TeX\ }

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User documentation is presented separately in \file{auguide.tex}.

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\changes{4.0a}{1998/01/16}{Initial version}
\changes{4.0a}{1998/01/31}{Move after process options, so \cs{clearpage} not in scope of twocolumn}
\changes{4.0a}{1998/01/31}{Rearrange the ordering so numerical ones come first. AO: David, what does this mean?}
\changes{4.0a}{1998/01/31}{use font-dependent spacing}
\changes{4.0a}{1998/01/31}{4.0d had twoside option setting twoside switch to false}
\changes{4.0a}{1998/01/31}{Move after process options, so the following test works}
\changes{4.0a}{1998/01/31}{print homepage}
\changes{4.0a}{1998/01/31}{protect against hyperref revtex kludges which are not needed now}
\changes{4.0a}{1998/06/10}{multiple preprint commands}
\changes{4.0a}{1998/06/10}{comma not space between email and homepage}
\changes{4.0a}{1998/06/10}{single space footnotes}
\changes{4.0b}{1999/06/20}{First modifications by Arthur Ogawa (mailto:arthur\_ogawa at sbcglobal dot net)}
\changes{4.0b}{1999/06/20}{Added localization of \cs{figuresname}}
\changes{4.0b}{1999/06/20}{Added localization of \cs{tablesname}}
\changes{4.0b}{1999/06/20}{AO: all code for \protect\classoption{10pt} is in this module.}
\changes{4.0b}{1999/06/20}{AO: all code for \protect\classoption{11pt} is in this module.}
\changes{4.0b}{1999/06/20}{AO: all code for \protect\classoption{12pt} is in this module.}
\changes{4.0b}{1999/06/20}{AO: made aps.rtx part of revtex4.dtx}
\changes{4.0b}{1999/06/20}{AO: remove duplicates}
\changes{4.0b}{1999/06/20}{call \cs{print@floats}}
\changes{4.0b}{1999/06/20}{Defer assignment until \cs{AtBeginDocument} time.}
\changes{4.0b}{1999/06/20}{Defer decision until \cs{AtBeginDocument} time}
\changes{4.0b}{1999/06/20}{Define three separate environments, defer assignment to \cs{AtBeginDocument} time.}
\changes{4.0b}{1999/06/20}{Frank Mittelbach, has stated in \protect\classname{multicol}: ``The kernel command \cs{@footnotetext} should not be modified.'' Thus, I have removed David Carlisle's redefinition of that command. Note, however, that later versions of \protect\classname{multicol} do not require this workaround. Belt and suspenders.}%
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\changes{4.0b}{1999/06/20}{only execute if appropriate}
\changes{4.0b}{1999/06/20}{Processing delayed to \cs{AtBeginDocument} time}
\changes{4.0b}{1999/06/20}{Removed invocation of nonexistent class option \protect\classoption{groupauthors} and all other class options that should only be invoked by the document. (Otherwise precedence of class options does not work.)}
\changes{4.0b}{1999/06/20}{Restore all media size class option of \protect\file{classes.dtx}}
\changes{4.0b}{1999/06/20}{Stack \cs{preprint} args flush right at right margin.}
\changes{4.0c}{1999/11/13}{(AO, 115) If three or more preprints specified, set on single line, with commas.}
\changes{4.0c}{1999/11/13}{(AO, 129) section* within appendix was producing appendixname}
\changes{4.0c}{1999/11/13}{*-form mandates pagebreak}
\changes{4.0c}{1999/11/13}{also spelled ``acknowledgements''.}
\changes{4.0c}{1999/11/13}{Do not put by REVTeX in every page foot}
\changes{4.0c}{1999/11/13}{grid changes via ltxgrid procedures}
\changes{4.0c}{1999/11/13}{grid changes with ltxgrid}
\changes{4.0c}{1999/11/13}{Insert procedure \cs{checkindate}}
\changes{4.0c}{1999/11/13}{Lose compatability mode.}
\changes{4.0c}{1999/11/13}{New ltxgrid-based  code, other bug fixes}
\changes{4.0c}{1999/11/13}{New option ``checkin''}
\changes{4.0c}{1999/11/13}{Prevent an inner footnote from performing twice}
\changes{4.0d}{2000/04/10}{Also alter how lists get indented.}
\changes{4.0d}{2000/04/10}{eprint takes an optional argument, syntactical only in this case.}
\changes{4.0d}{2000/04/10}{New option}
\changes{4.0d}{2000/05/10}{More features and bug fixes: compatability with longtable and array packages. Now certainly incompatible with multicol.}
\changes{4.0d}{2000/05/17}{make longtable trigger the head, too}
\changes{4.0d}{2000/05/18}{But alternative spelling is deprecated.}
\changes{4.0e}{2000/09/20}{New option showkeys}
\changes{4.0e}{2000/11/14}{Bug fixes and minor new features: title block affiliations can have ancillary data, just like authors; clearpage processing revamped, with floats staying in order; widetext ornaments.}
\changes{4.0e}{2000/11/21}{adornments above and below.}
\changes{4.0f}{2001/02/13}{Last bug fixes before release.}
\changes{4.0rc1}{2001/06/17}{Running headers always as if two-sided}
\changes{4.0rc1}{2001/06/18}{grid changes with push and pop}
\changes{4.0rc1}{2001/06/18}{grid changes with push and pop}
\changes{4.0rc4}{2001/07/23}{hyperref is no longer loaded via class option: use a usepackage statement instead}
\changes{4.1a}{2008/01/18}{(AO, 457) Endnotes to be sorted in with numerical citations.}%
\changes{4.1a}{2008/01/18}{(AO, 451) ``Cannot have more than 256 cites in a document''}%
\changes{4.1a}{2008/01/18}{(AO, 457) Endnotes to be sorted in with numerical citations.}%
\changes{4.1a}{2008/01/18}{(AO, 460) ``Proper style is "FIG. 1. ..." (no colon)''}%
\changes{4.1a}{2008/01/18}{(AO, 478) \cs{ds@letterpaper}, so that ``letterpaper really is the default''}%
\changes{4.1a}{2008/01/18}{(AO, 488) Change processing of options to allow an unused option to specify society and journal}%
\changes{4.1a}{2008/01/19}{(AO, 461) Change the csname revtex uses from @dotsep to ltxu@dotsep. The former is understood in mu. (What we wanted was a dimension.)}%
\changes{4.1a}{2008/01/19}{For natbib versions before 8.21, \cs{NAT@sort} was consulted only as natbib was being read in. Now it is fully dynamic.}
\changes{4.1b}{2008/05/29}{The csname substyle@ext is now defined without a dot (.), to be compatible with \LaTeX usage (see @clsextension and @pkgextension).}
\changes{4.1b}{2008/06/01}{(AO) Implement bibnotes through \cs{frontmatter@footnote@produce} instead of \cs{bibnotes@sw}}%
\changes{4.1b}{2008/06/01}{Add option reprint, opposite of preprint, and preferred alternative to twocolumn}
\changes{4.1b}{2008/06/29}{(AO, 455) Be nice to a list within the abstract (assign \cs{@totalleftmargin}).}
\changes{4.1b}{2008/06/30}{(AO) Structure the Abstract using the \texttt{bibliography} environment}
\changes{4.1b}{2008/07/01}{(AO) coordinate \cs{if@twoside} with \cs{twoside@sw}}
\changes{4.1b}{2008/07/01}{(AO) make settings at class time instead of deferring them to later.}
\changes{4.1b}{2008/07/01}{(AO) No longer need to test \cs{chapter} as of \texttt{natbib} version 8.2}
\changes{4.1b}{2008/07/01}{(AO) No longer use \cs{secnumarabic@sw}, instead use \cs{setup@secnums}}
\changes{4.1b}{2008/07/01}{(AO) Provide more diagnostics when \cs{@society} is assigned.}
\changes{4.1b}{2008/07/01}{(AO) provide option longbibliography}
\changes{4.1b}{2008/07/01}{Add \cs{@hangfroms@section}}
\changes{4.1b}{2008/07/01}{Break out \cs{@caption@fignum@sep}}
\changes{4.1b}{2008/07/01}{Class option galley sets \cs{preprintsty@sw} to false}
\changes{4.1b}{2008/07/01}{Code relating to new syntax for frontmatter has been placed in \file{ltxfront.dtx}}
\changes{4.1b}{2008/07/01}{Package textcase is now simply a required package}
\changes{4.1b}{2008/07/01}{Procedures \cs{@parse@class@options@society} and \cs{@parse@class@options@journal} and friends}
\changes{4.1b}{2008/07/01}{Read in all required packages together}
\changes{4.1b}{2008/07/01}{Remove options newabstract and oldabstract}
\changes{4.1b}{2008/08/01}{Section numbering via procedures \cs{secnums@rtx} and \cs{secnums@arabic}.}
\changes{4.1b}{2008/08/04}{As with author formatting, rag the right more, and assign \cs{@totalleftmargin}. Also neutralize \cs{def@after@address}.}%
\changes{4.1b}{2008/08/04}{Rag the right even more: .8\cs{hsize}. Also, assign \cs{@totalleftmargin}.}%
\changes{4.1b}{2008/08/04}{The \texttt{rmp} journal substyle selects \texttt{groupedaddress} by default.}%
\changes{4.1b}{2008/08/04}{Use \cs{setup@hook} to initialize all.}
\changes{4.1c}{2008/08/15}{Document class option longbibliography via \cs{substyle@post}}
\changes{4.1d}{2009/03/27}{Definition of \cs{ @fnsymbol} follows fixltx2e.sty}
\changes{4.1e}{2008/06/29}{(AO, 455) be nice to a list within the abstract}
\changes{4.1f}{2009/07/07}{(AO, 513) Add class option linenumbers: number the lines a la \classname{lineno}}
\changes{4.1f}{2009/07/07}{(AO, 516) Merged references are separated with a semicolon}
\changes{4.1f}{2009/07/10}{(AO, 520) Automatically produce \cs{bibliography} command when needed}%
\changes{4.1f}{2009/07/11}{(AO, 521) Lonely bibliography head}%
\changes{4.1f}{2009/07/11}{(AO, 522) Warn if software is expired}%
\changes{4.1f}{2009/07/15}{(AO, 523) Add class option nomerge, to turn off new natbib 8.3 syntax}
\changes{4.1f}{2009/07/20}{(AO, 524) Makes no sense if citations are superscript numbers and so are footnotes}
\changes{4.1f}{2009/10/05}{(AO, 530) \cs{@fnsymbol}: Failed to import fixltx2e.sty technology. Return to LaTeX core.}
\changes{4.1g}{2009/10/07}{(AO, 525) Remove phantom paragraph above display math that is given in vertical mode}%
\changes{4.1g}{2009/10/07}{(AO, 538) \cs{MakeTextUppercase} inappropriately expands the double backslash}
\changes{4.1h}{2009/10/09}{(AO) Remove expiry code in the release software}%
\changes{4.1i}{2009/10/23}{(AO, 541) Defer assignment of \cs{cite} until after natbib loads}
\changes{4.1j}{2009/10/24}{(AO, 549) Repairing natbib's \cs{BibitemShut} and \cs{bibAnnote}}
\changes{4.1j}{2009/10/25}{(AO, 545) hypertext capabilities off by default; enable with \classoption{hypertext}}
\changes{4.1j}{2009/10/25}{(AO, 552) Repair spacing in \cs{onlinecite}}
\changes{4.1k}{2009/11/06}{(AO, 554) give the \cs{newlabel} command syntax appropriate to the hyperref package}
\changes{4.1n}{2009/11/06}{(AO, 565) restore 4.0 behavior: invoking class option preprint implies class option preprintnumbers}
\changes{4.1n}{2009/11/30}{(AO, 566) restore 4.0 behavior: flush column bottoms}
\changes{4.1n}{2009/12/05}{(AO, 569) Use of \classname{hyperref} interferes with column balancing of last page}%
\changes{4.1n}{2009/12/09}{(AO, 569) execute the after-last-shipout procedures from within the safety of the output routine}%
\changes{4.1n}{2010/01/02}{(AO, 571) Interface \cs{set@footnotewidth} for determining the set width of footnotes}%
\changes{4.1n}{2010/01/02}{(AO, 572) Independent footnote counter for title block. Abstract footnote counter shared with body.}%
\changes{4.1n}{2009/12/13}{(AO, 573) arrange to load \classname{lineno} after any other packages.}%
\changes{4.1n}{2010/01/04}{(AO, 575) the default for journal prstper is longbibliography}%
\changes{4.1n}{2010/01/04}{(AO, 576) In .bst files, remove support for the annote field}%
\changes{4.1n}{2010/01/02}{(AO) fine-tune spacing above and below widetext}%
\changes{4.1n}{2010/01/02}{(AO, 571) class file must set \cs{splittopskip}; fine tune \cs{skip}\cs{footins}; \cs{footnoterule} defined in terms of \cs{skip}\cs{footins}}%
\changes{4.1n}{2010/01/02}{(AO, 572) \cs{@makefntext} and \cs{frontmatter@makefntext} must be defined harmoniously}%
\changes{4.1o}{2010/02/02}{(AO, 575) Automatically incorporate the (Bib\TeX-generated) .bbl into an explicit \env{thebibliography}}%
\changes{4.1o}{2010/02/05}{(AO, 549) Remove patch to natbib, which is now at version 8.31a}
\changes{4.1o}{2010/02/07}{(AO, 578) accommodate the possible space character preceding \cs{BibitemShut}.}
\changes{4.1o}{2010/02/05}{(AO, 579) Endnote shall comprise their own Bib\TeX\ entry type: @FOOTNOTE.}
\changes{4.1o}{2010/02/10}{(AO, 580) Provide a document class option to turn off production of eprint field in bibliography.}
\changes{4.1o}{2010/02/12}{(AO, 580) Control .bst at run time.}%
\changes{4.1o}{2010/02/09}{(AO, 581) Handle case: merged references, with first ending in a stop character.}
\changes{4.1p}{2010/02/24}{(AO, 583) Provide interface to \classname{ltxgrid} \cs{onecolumn@grid@setup} and \cs{twocolumn@grid@setup}}
\changes{4.1p}{2010/02/24}{(AO, 584) Per MD, remove trailing space character from each journal abbreviation: it had caused an extraneous space in the .bbl}
\changes{4.1q}{2010/04/01}{(AO, 586) When .bbl is pasted into the document, prevent automatic bibliography inclusion.}%
\changes{4.1q}{2010/04/13}{(AO, 588) Only write REV\TeX\ -specific BibTeX .bib data if the .bst style is set by REVTeX.}%
\changes{4.1r}{2010/06/22}{(AO, 595) Provide \cs{lovname} along with other List of Videos definitions.}%

\iffalse ltxdoc klootch
This file has version number 4.1r, last revised 2010/07/25/20:33:00.\fi

\section{Introduction}

This is the author's guide to the AIP substyles for REV\TeX\ ~4.1,
providing a useful formatting tool
for \LaTeX\ users submitting papers to journals
published by the American Institute of Physics.
This guide is intended as an adjunct to the documentation for REV\TeX\ \ itself
(published by the American Physical Society),
so information contained therein is not repeated here,
except as it bears on the specific features of the AIP substyles.

\subsection{Prerequisite Documentation}

The following documentation should be considered your first source of information
on how to prepare your document for use with this format;
they are to be found within the APS REV\TeX\ ~4.1 distribution.
Updated versions of these are maintained at
the REV\TeX\ ~4.1 homepage located at \url{http://authors.aps.org/revtex4/},
are also available at the Comprehensive \TeX\ Archive Network (CTAN, see \url{http://www.ctan.org/}),
and form part of the \TeX Live distribution of \TeX.
\begin{itemize}
\item \textit{Author's Guide to REV\TeX\ ~4.1}
\item \textit{REV\TeX\ ~4.1 Command and Options Summary}
\item \textit{What's New in REV\TeX\ ~4.1}
\end{itemize}
The present guide builds upon these documents, with which you should already be familiar.

The AIP substyles distribution for REV\TeX\ ~4.1 includes
a sample document (\file{aipsamp.tex}),
a good starting point for
the manuscript you are preparing for submission to an AIP journal.

By using REV\TeX\ 's \textit{Author's Guide to REV\TeX\ ~4.1}, you can develop your
document until it contains all of the content you desire.
This guide informs you on document class options, commands, and
markup guidelines specific to AIP journals.

\subsection{Software Requirements}

This guide assumes a working REV\TeX\ ~4.1 installation including the AIP substyles.
Please see the installation guide included with the distribution.\cite{Note1}

Please note that the AIP substyles work {\it only} with REV\TeX\ ~4.1:
the original REV\TeX\ ~4.0 release does {\it not} make the AIP substyle available, nor is it compatible with them.

For your computer to run REV\TeX\ ~4.1 with the AIP substyles, the following are required:
\begin{itemize}
\item
a working installation of \LaTeX\,
\item
REV\TeX\ ~4.1 and all packages it requires,
\item
the AIP substyles for REV\TeX\ ~4.1, and
\item
any further \LaTeX\ packages used in your document.
\end{itemize}

The easiest way to obtain all of the needed software is to install an up-to-date distribution of \TeX,
like \TeX Live, available on CTAN.

To obtain the most up-to-date version of this software, please see \url{http://www.aip.org/pubservs/compuscript.html}.

\subsection{Submitting to AIP Journals}

Authors preparing a manuscript for submission to
AIP journals should consult the Information for Contributors for the applicable journal,
available through links at \url{http://journals.aip.org/}.
These requirements are not covered systematically in this author's guide;
you are responsible for understanding the requirements of the particular journal to which
you will submit your article.

For further information about journal requirements, contact the Editorial
Office of the appropriate journal. (Follow links at \url{http://journals.aip.org/}.)

\subsection{Contact Information}\label{sec:resources}%
Any bugs, problems, or inconsistencies concerning the AIP journal substyles
should be reported to AIP support at \href{mailto:tex@aip.org}{tex@aip.org}.
Reports should include information on the error and a
\textit{small} sample document that manifests the problem, if possible.
(Please don't send large files!)

Feedback concerning REV\TeX\ ~4.1 itself should be sent, as usual,
to the American Physical Society at\\ \href{mailto:revtex@aps.org}{revtex@aps.org}.

To determine if the problem you are experiencing belongs to REV\TeX\ \ or is specific to the
AIP substyles, simply remove \texttt{aip} from your document class options and rerun
your document. If the problem goes away, you may assume that it is due to the AIP substyles;
if not, it belongs to REV\TeX\ .

\section{Sample \LaTeXe\ Document}
As the REV\TeX\ \ documentation makes clear, your document employs a \LaTeXe\ document class
(specifically \texttt{revtex4-1.cls}), so you should use
the \LaTeXe\ commands and environments familiar to you with, say, the
standard article class \texttt{revtex4-1.cls}, and you will be able to
employ many of the packages you are used to using with \LaTeXe.

Using \texttt{aipsamp.tex} as an example,
your document will start with the usual REV\TeX\ \ \cmd\documentclass\ statement, but with
a particular document class option \texttt{aip} that specifies the AIP substyle:
\begin{verbatim}
\documentclass[aip]{revtex4-1}
\end{verbatim}
You will then invoke the \LaTeXe\-compatible packages your document requires, say:
\begin{verbatim}
\usepackage{graphicx}%
\usepackage{dcolumn}%
\usepackage{bm}%
\end{verbatim}
follow up with your document content:
\begin{verbatim}
\begin{document}
...
\end{verbatim}
and finish with a statement specifying your Bib\TeX\ database:
\begin{verbatim}

\section{Introduction}
Articles published in American Physical Society journals are converted to
an XML file during final journal production. Other formats such
as PDF are derived directly from the XML, which constitutes the version of record.
Even before journal production, the APS editorial process can make use
of the information in a properly prepared manuscript. Information such
as title, authors, affiliations, PACS codes, etc., can be automatically
extracted and used to populate our manuscript database. References can
also be culled, cross-checked for accuracy, and used to create a
linked version for referees and editors. Moreover, time can be saved
as referrals can be made electronically rather than by conventional
mail. Thus, a well-prepared electronic manuscript can enhance the
entire peer review process from author to reader while making the
whole process less expensive. To this end, authors should follow the
guidelines in this document when preparing their submissions to \textit{Physical Review Letters},
 \textit{Reviews of Modern Physics},  \textit{Physical Review A-E},  \textit{Physical Review Special Topics - Accelerators and Beams}, and  \textit{Physical Review Special Topics - Physics Education Research}.

Updated versions of this document will be made available at  \url{http://authors.aps.org/revtex4/}. For more complete
descriptions of how to use the REV\TeX\ \ 4.1 macros, please see the
\textit{REV\TeX\ ~4.1 Author's Guide} included with the REV\TeX\ ~4.1
distribution. Questions about REV\TeX\ \ 4.1 and using it to submit to APS journals may be
emailed to \texttt{revtex@aps.org}.

\section{Formatting}
\subsection{Preprint, reprint, and twocolumn options}
REV\TeX\ ~4.1 offers a new \classoption{reprint} class option to typeset a manuscript
in a format that is a close approximation to the actual journal's appearance. It should
be emphasized that this is only an \textit{approximation}; a manuscript may be substantially different
in length or appearance after it goes through our production process. This is mostly due to the choice
of fonts and the scaling of figures.

REV\TeX\ \ 4.1 is designed to
make it straightforward to switch between two-column and single-column
formatting just by changing the class option. Authors may submit with
either the \classoption{reprint} or the \classoption{twocolumn} class options.
The \classoption{preprint} primarily does three things: It increases
the font size to 12pt, increases the line spacing, and changes the
formatting to single column.

\subsection{Paper size}
Manuscripts should be submitted to APS formatted for letter size
paper. Papers are sent electronically to referees who may
want to print them out. Letter size formatting ensures that this will
be trouble free for all referees.

\section{Marking up front matter}
Perhaps the most important macros are those
pertaining to the markup of the front matter (title, authors,
affiliations, abstract, PACS codes, etc.). Note that proper
use of the REV\TeX\ \ 4.1 macros means that explicit centering environments
in the front matter are not needed and should not be used.

\subsection{Title}
The title of the manuscript should be specified using the \m{title} macro. A
double backslash {\textbackslash\textbackslash} may be used to force a line break in a long
title.

\subsection{Authors, affiliations, and collaborations}
\label{sec:authors}
REV\TeX\ \ 4.1 makes it straightforward to markup up author names and link them up properly with affiliations. Authors should let REV\TeX\ \ 4.1 do the work of grouping authors and affiliations and, if using the superscript style, numbering affiliations. Please follow these guidelines:
\begin{itemize}
\item Use a single \m{author} macro for each author's name. REV\TeX\ \ 4.1 automatically puts in all commas and the word `and.'
\item Use the \m{surname} macro to explicitly indicate if an author's family name consists of more than one name or if the family name is not the author's last name.
\item The \m{email} macro may be used to specify an author's e-mail
address. The \m{thanks} macro must not be used for this. Only the
e-mail address itself may appear in the macro's required argument.
\item The \m{homepage} macro may be used to specify a URL associated
with an author. The \m{thanks} macro must not be used for this. Only the
URL may appear in the macro's required argument.
\item The \m{altaffiliation} macro may be used to specify an alternate
affiliation or temporary address for an author. The \m{thanks} macro
must not be used for this. Only the affiliation
may appear in the macro's required argument.
\item The \m{thanks} macro may be used only if one of the more
specific macros list above does not apply.
\item Use a single \m{affiliation} for each affiliation.
\item Superscripts linking authors to affiliations must be
accomplished using the \classoption{superscriptaddress} class option
rather than putting in explicit superscripts by hand.
\item A collaboration may be specified by using the \m{collaboration}
macro. The \m{author} macro must not be used for collaborations.
\end{itemize}
\subsection{Abstract}
The abstract must be specified using the \env{abstract}
environment. Note that in REV\TeX\ \ 4.1, the abstract must come before
the \m{maketitle} command. REV\TeX\ \ 4.1 now allows the the use of the \env{description}
environment within the abstract to provide \textit{structured abstracts}. For instance, \textit{Physical Review C} would like authors to provide abstracts with sections summarizing the paper's  \textbf{Background}, \textbf{Purpose}, \textbf{Method}, \textbf{Results}, and \textbf{Conclusions}. This can be accomplished in the following manner:
\begin{verbatim}
\begin{abstract}
\begin{description}
\item[Background] This part would describe the
context needed to understand what the paper
is about.
\item[Purpose] This part would state the purpose
of the present paper.
\item[Method] This part describe the methods
used in the paper.
\item[Results] This part would summarize the
results.
\item[Conclusions] This part would state the
conclusions of the paper.
\end{description}
\end{abstract}
\end{verbatim}

\subsection{PACS codes}
If supplying suggested PACS codes, they must be supplied as a
comma-separated list using a single \m{pacs} macro.

\section{References and footnotes}
Authors are strongly encouraged
to use Bib\TeX\ when preparing their bibliographies.  If Bib\TeX\ is used, current production processes
require that the \texttt{.bbl} file be included directly into the
manuscript's main \texttt{.tex} file. REV\TeX\ \ 4.1 comes with two Bib\TeX\ style files for formatting
references, one for the \textit{Physical Review} journals and one
for \textit{Review of Modern Physics}. The Bib\TeX\ styles support the optional display of journal article titles in the bibliography. To display the titles, use the \texttt{longbibliography} class option.

The following apply whether
Bib\TeX\ is used or not.
\begin{itemize}
\item Authors should use the \m{cite} and \m{bibitem} commands to create
bibliographies and to refer to items in the bibliography. ``By hand"
numbering of references should be avoided.
\item REV\TeX\ ~4.1 provides new syntax for combining multiple citations into a single entry in the bibliography and for putting extra text before and after a reference. Please refer to \textit{REV\TeX\ ~4.1 Author's Guide} included with the REV\TeX\ ~4.1 distribution for full details.
\item Footnotes must be specified using the \m{footnote}
macro. REV\TeX\ \ 4.1 will place the footnotes in
the bibliography for the \textit{Physical Review}
journals. Please note that even if you don't use Bib\TeX, you may have to run Bib\TeX\ to get the footnotes to appear. Footnotes giving additional information about authors (such
as e-mail addresses) must not be specified using the \m{footnote}
macro (see Section~\ref{sec:authors}).
\item Avoid custom footnotes using \m{footnotemark} and \m{footnotetext} [except in the context of tables (see
Section~\ref{sec:tablenotes})].
\item References should be formatted and specified according to the
\textit{Physical Review Style Guide}. Note that using Bib\TeX\ automatically ensures this.
\item In journal styles that use superscript citations (\textit{e.g.}, Phys.~Rev.~B), use \m{onlinecite} to create references that appear on the baseline (\textit{i.e.}, Ref.~4). For superscript citations, use the proper journal class option  rather than creating superscripts by hand.
\item URLs should be specified using the \m{url} macro. Bib\TeX\ will automatically take
care of this if the \texttt{url} field is used.
\item E-print identifiers should be included using the \m{eprint} macro. Bib\TeX\ will automatically take care of this if the \texttt{eprint} field is used.
\end{itemize}

\section{Body of the paper}
\subsection{Sectioning and cross-referencing}
For sectioning a manuscript, the basic rule is to use the appropriate
sectioning commands (\m{section}, \m{subsection}, \m{subsubsection},
\textit{etc.}). Cross-referencing a section must be done by using the
proper \m{label} and \m{ref} commands. Cross-referencing by hand is
not allowed. \m{part}, \m{chapter}, and \m{subparagraph} should not be
used.

\subsection{Appendices}
Appendices should be specified using the \m{appendix} command which
specifies that all following sections create with the \m{section}
commands are appendices. If there is only one appendix, then the
\m{appendix*} command should be used instead.

\subsection{Acknowledgments}
Any acknowledgments should be included by using the
\env{acknowledgments} environment. Note that in REV\TeX\ ~4.1, this is
an environment and not a command.

\subsection{Counters}
No counters may be created and the standard ones may not be
altered. If an exceptional label is needed for an equation, the \m{tag}
command (requires the \classoption{amsmath} class option) should be used. Please
note that the use of the \m{tag} command may conflict with the use of the \classoption{hyperref} package
due an incompatibility between \classoption{amsmath} and \classoption{hyperref}.

\subsection{Fonts}
It is preferable to avoid the older \TeX\ and \LaTeX\ 2.09 macros for
controlling fonts such as \m{rm}, \m{it}, \textit{etc.} Rather, it is
better to use the macros introduced in \LaTeXe.  If the older font
commands are used (they really should be avoided!), be sure to use
curly braces to properly limit the extent of the font
change. \verb+{\bf ...}+ is the correct method.
Commands for controlling text and math font changes are summarized in
Table~\ref{tab:fonts}.

\begin{table}
\caption{\label{tab:fonts}\LaTeXe\ and AMS-\LaTeX\ font summary.}
\begin{ruledtabular}
\begin{tabular}{lp{2in}}
\m{textit} & Italics. Replaces \m{it}\\
\m{textbf} & Bold face. Replaces \m{bf}\\
\m{textrm} & Roman. Replaces \m{rm}\\
\m{textsl} & Slanted. Replaces \m{sl}\\
\m{textsc} & Small caps. Replaces \m{sc}\\
\m{textsf} & Sans serif. Replaces \m{sf}\\
\m{texttt} & Typewriter. Replaces \m{tt}\\
\m{textmd} & Medium series\\
\m{textnormal} & Normal\\
\m{textup} & Upright\\
\m{mathbf} & Bold face\\
\m{mathcal} & Replaces \m{cal}\\
\m{mathit} & Italics\\
\m{mathnormal} & Replaces \m{mit}\\
\m{mathsf} & Sans serif\\
\m{mathtt} & Typewriter\\
\m{mathfrak} & Fraktur: Requires \classoption{amsfonts} or \classoption{amssymb} class option\\
\m{mathbb} & Bold blackboard: Requires \classoption{amsfonts} or \classoption{amssymb} class option\\
\m{bm} & Bold Greek and other math symbols: Requires
\verb+\usepackage{bm}+ and may require the \classoption{amsfonts} class
option
\end{tabular}
\end{ruledtabular}
\end{table}

Bold Greek letters and other bold math symbols should be accomplished
with the use of \texttt{bm.sty} which is distributed as a required
tool with the latest versions of \LaTeXe\ and should be loaded via
\verb+\usepackage{bm}+. This package introduces the \m{bm}
macro. Some bold characters may require using the
\classoption{amsfonts} class option.

New fonts may not be declared with \m{newfont}. Font attribute
commands for selecting a font family, shape, and series are all
disallowed; the standard \LaTeXe\ font selection macros list above
should be used instead.

Finally, the \m{symbol} macro is also not allowed.

\subsection{Environments}
\subsubsection{Lists}
The standard list environments \texttt{itemize}, \texttt{enumerate},
and \texttt{description} are allowed. The \m{item} macro with or without
the optional argument is also allowed. Customization of the list environments
(with macros such as \m{labelstyle}, \m{labelitemi}, \m{labelenumi},
\m{itemsep}, etc.) is allowed but may be ignored in production.
Generalized lists (\m{begin\{list\}}) and trivial lists
(\m{begin\{trivlist\}}) are not allowed.

\subsubsection{Other Environments}
Creating generalized new environments with \m{newenvironment} is not
allowed. Creating a new theorem environment with \m{newtheorem} is
allowed though.

The tabbing environment and the macros \m{=}, \m{$>$}, \m{`}, and
\m{'} are allowed but may be ignored in production. Conversion
programs used in production should recognize the escapes \m{a=},
\m{a'}, and \m{a`} for using the corresponding accents within a
tabbing environment though.

The \env{verbatim} environment is allowed.

\subsection{Boxes}
Most boxes and macros to manipulate them are not allowed. These
include \m{raisebox}, \m{parbox}, \m{minipage}, \m{rulebox},
\m{framebox}, \m{mbox}, \m{fbox}, \m{savebox}, \m{newsavebox},
\m{sbox}, \m{usebox}, and the environment \m{begin\{lrbox\}}. Rules
produced with \m{rule} are not allowed.

\subsubsection{Margin Notes}
Margin notes created with \m{marginpar} are not allowed, as are the
associated style parameters \m{marginparwidth}, \m{marginparsep}, and
\m{marginparpush}.

\section{Math Markup}
In general, all math markup and the standard math environments from
\LaTeXe\ are allowed. These include \m{begin\{math\}},
\m{begin\{displaymath\}}, \m{begin\{equation\}},
\m{begin\{eqnarray\}}, and \m{begin\{eqnarray*\}}. The shortcuts \$,
\$\$, \m{[}, and \m{]} are allowed. In addition, authors may use
almost all of the additional markup introduced by AMS-\LaTeX\ by using
the \classoption{amsmath} class option. The explicit exceptions are
\m{genfrac}, \m{boxed}, and \m{smash}. The markup contained in
\texttt{amsextra} and \texttt{amsthm} may not be used
though. Commutative diagrams created with the \texttt{amscd} package
are acceptable.

\section{Figures}
\subsection{Figure inclusions}
Figures should be included into a REV\TeX\ ~4.1 manuscript by using the
standard \LaTeXe\ macros. \LaTeXe\ includes
several powerful packages for including the files in various
formats. The two main packages are \texttt{graphics} and
\texttt{graphicx}. Both offer a macro called
\m{includegraphics};
they mainly differ in how arguments for
controlling figure placement (\textit{e.g.}, scaling and rotation)
are passed to the \m{includegraphics}.

The \env{figure} environment should be used to add a caption to the
figure and to allow \LaTeX\ to number and place the figures where they
fit best.  If a figure needs to be referred to in the text,
rather than manually numbering the figures a \m{label} should be added
to the figure environment (best practice is to put the label within
the argument of the \m{caption} command) and the \m{ref} macro should be used to
reference this label. Figures that span the page should use the
\m{figure*} environment. The \env{picture} environment must not be
used directly (one can include an Encapsulated PostScript figure that
was produced using the \env{picture} environment of course).

\subsection{\label{sec:figplace}Figure placement}
Figures should be placed as close as possible to the point where they are first
referenced. There is no need to place all figures
separately at the end of the manuscript and it is preferred that
authors leave the figures in their natural locations. Authors may
also find useful the REV\TeX\ ~4.1 \classoption{floatfix} class option
which adds emergency float placement processing to avoid ``stuck''
floats which would otherwise be deferred to the end of the job (and
can lead to the fatal \texttt{``Too many unprocessed floats''}
message).

\section{Tables}
\label{sec:tables}
The standard \LaTeXe\ table formatting environments are supported as is
the use of the \texttt{longtable} package. Tables may be reformatted
during production to meet APS style guidelines.
Here are some helpful hints for trying to get tables formatted correctly:
\begin{itemize}
\item Use the \texttt{longtable} package to get tables to break
across pages.
\item The macro \m{squeezetable} will reduce the font size of the
table. This macro must occur within a group outside the table
environment. The proper markup is:
\begin{verbatim}
\begingroup
\squeezetable
\begin{table}
...
\end{table}
\endgroup
\end{verbatim}
\item Try using the float placement option \texttt{H} which will
enable \LaTeX\ to break a float across pages. Long tables are more
attractively set with \env{longtable} however.
\begin{verbatim}
\begin{table}[H]
\begin{ruledtabular}
\begin{tabular}
...
\end{tabular}
\end{ruledtabular}
\end{table}
\end{verbatim}
\end{itemize}

\subsection{Doubled rules and table formatting}
REV\TeX\ \ 4.1 provides the \env{ruledtabular} environment which
automatically puts the scotch rules (double lines) around tables and
formats all enclosed \env{tabular} environments to the full width of
the tables and improves inter-column spacing. This environment should
be used whenever possible.

\subsection{Wide tables}
When typesetting using \classoption{twocolumn}, tables can either span
a single column or both columns. Using the '\verb+*+'-ed version of
the \env{table} or \env{longtable} environments produces wide tables
that span the columns.

Tables that are very wide and that may be better typeset in a
landscape orientation (rotated 90 degrees) should be enclosed in a
\env{turnpage} environment. This will place the rotated table on its own
page. Note that some dvi previewers may not be able to show the table
properly, but \texttt{dvips} and \texttt{pdflatex} work correctly.

\subsection{Table placement}
Tables should be placed as close as possible to the point where they
are first referenced. There is no need to place all tables separately
at the end of the manuscript and this is not desirable for APS
purposes. The class option \classoption{floatfix} may be helpful for
table placement as well as figure placement (see Section~\ref{sec:figplace}).

\subsection{Aligning columns on a decimal point}
The standard \LaTeXe\ macro package \classoption{dcolumn} should be
used to accomplish this.

\subsection{Tablenotes}
\label{sec:tablenotes}
Footnotes in tables (tablenotes) should use the \m{footnote}
macro. However, if more than one reference to the same footnote is
needed, authors may use \m{footnotetext} and \m{footnotemark}. This
will produce notes (labeled by lower-case roman letters) inserted
below the table rather than in the reference section or at the bottom
of the page.

\section{Author-defined macros}
Authors may define convenience macros to save keystrokes. This means
that the macros may not invoke \TeX\ macros such as \m{if} or other
context dependent commands. Also, \LaTeXe\ provides three macros for
declaring new commands: \m{providecommand}, \m{newcommand}, and
\m{renewcommand} (as well as their `\verb+*+'-ed versions). These
should be used. Authors may not use \TeX\relax's low-level commands
\m{def}, \m{edef}, and \m{gdef}.

\section{Summary}
To ensure the best use of \TeX\ manuscripts, authors need to follow
the guidelines specified here. Use of low-level formatting commands to
finely control horizontal and vertical spacing may be ignored during
production, or even worse, make it impossible to convert the
manuscript to XML. Authors should keep
things as simple as possible and correctly use the proper REV\TeX\ ~4.1
or \LaTeXe\ macros. Any questions about usage may be directed to
\texttt{revtex@aps.org}.

\end{document}

\section{Introduction}

This is the author's guide to REV\TeX\ ~4.1, the preferred submission
format for all APS and AIP journals. This guide is intended to be a concise
introduction to REV\TeX\ ~4.1. The documentation has been separated out
into smaller units to make it easier to locate essential
information.

The following documentation is also part of the REV\TeX\ ~4.1
distribution. Updated versions of these will be maintained at
the REV\TeX\ ~4.1 homepage located at \url{http://authors.aps.org/revtex4/}.
\begin{itemize}
\item \textit{APS Author Guide for REV\TeX\ ~4.1}
\item \textit{Author's Guide to AIP Substyles for REV\TeX\ ~4.1}
\item \textit{REV\TeX\ ~4.1 Command and Options Summary}
\item \textit{What's New in  REV\TeX\ ~4.1}
\end{itemize}
This guide assumes a working REV\TeX\ ~4.1
installation. Please see the installation instructions included with the
distribution.
\subsection{Changes in REV\TeX\ ~4.1}
The REV\TeX\ \ system for \LaTeX\ began its development in 1986 and has
gone through three major revisions since then.  REV\TeX\ ~4 was released in August, 2001. Since that time,
many user requests for new features were received. The main goals for  REV\TeX\ ~4.1 are to incorporate
this user feedback and provide support for the journals of the American Institute of Physics (AIP) . It incorporates the following changes:

\begin{itemize}
\item \textbf{Added support for APS journal \textit{Physical Review Special Topics -- Physics Education Research}}.
\item \textbf{Added support for AIP journals.} There is now an explicit \texttt{aip} society option along with support for AIP journals. Please see the \textit{Author's Guide to AIP Substyles for REV\TeX\ ~4.1}. In addition, REV\TeX\ ~4.1 provides an extensible system for the easy addition of new collections of journals.
\item \textbf{Endnotes now ordered correctly.} Endnotes in the bibliography now appear in the correct order, interleaved with citations.
\item \textbf{Multiple references in a single citation supported using a special starred (*) argument to the \cmd\cite\ command.} One of the major new features in 4.1 made possible by the joint work on \texttt{natbib 8.3}. Multiple Bib\TeX\ entries can be combined into a single \cmd\bibitem\ command.
\item \textbf{Free form text can be prepended and appended to a bibliographic entry using the special starred (*) argument to the \cmd\cite\ command.} Often a citation in the bibliography will have explanatory text such as \textit{See also} or \textit{and references therein} before and after the actual citation. The new REV\TeX\ ~4.1 \cmd\cite\ command allows the specification of both text to precede and follow a citation.
\item \textbf{Structured Abstracts.} Use of the \texttt{description} environment in abstracts now provides for ``structured" abstracts.
\item \textbf{Figures referring to videos now supported.} A ``figure" may now be labeled as a \textbf{Video} by using the \texttt{video} environment. A frame from the video may be included in the figure and a URL to link the caption's label to the online video also may be included. There is also a \cmd\listofvideos\ command.
\item \textbf{Better support for arXiv.org in Bib\TeX\ } Three more Bib\TeX\ fields have been added: \texttt{SLACcitation}, \texttt{archivePrefix}, and \texttt{primaryClass} in addition to the existing field \texttt{eprint}.
\item \textbf{Improved Bib\TeX\ \texttt{bst} files.} In addition to the new features above, numerous other improvements to the APS \texttt{bst} files have been made, including support for displaying journal article titles (using the new \texttt{longbibliography} option) and many fixes for \textit{Reviews of Modern Physics}. Also, long author lists are no longer automatically truncated.
\item \textbf{\cmd\footnote\ in \cmd\widetext\ and \texttt{table*} environments improved.} \cmd\footnote\relax s in  the \cmd\widetext\ or \texttt{table*} environments are now correctly placed and formatted.
\item \textbf{Email addresses no longer print twice on papers less than one page long.}
\item \textbf{\texttt{eqnarray} alignment improved.}
\item \textbf{\cmd\collaboration\ can be used with the \texttt{groupedaddress} option now.}
\item \textbf{\texttt{letterpaper} now ensured as default paper size.}
\item \textbf{Table of Contents formatting improved.}
\item \textbf{Support for the \texttt{longtable} package improved.}
\item \textbf{\texttt{reftest} restored.}
\item \textbf{Compatibility with the \texttt{geometry, lineno, lscape} and \texttt{colortbl} packages improved.} For line numbering, rather than using \texttt{lineno.sty} directly, the \texttt{linenumbers} class option should be used (this will call in \texttt{lineno.sty} with a proper set of default parameters).
\item \texttt{hyperref} \textbf{fixes}. Improvements were to make footnotes work better with the \texttt{hyperref} package. In particular, table footnotes were fixed. More anchors for \texttt{hyperref} were also added (titlepage, abstract, and acknowledgements).
\item \textbf{Documents can have more than 256 \cmd\cite\ commands now.}
\item \textbf{\cmd\listoffigures\ and \cmd\listoftables\ fixed.}
\item \textbf{Figure and table labels in captions now reflect proper APS style.}
\item \textbf{RMP style files conform better to RMP style guidelines.}
\item \textbf{Section heading upper-casing improved.}
\item \textbf{Repeated characters at start of affiliation no longer disappear when using \texttt{groupedaddress} option.}
\item \textbf{There have been many other bug fixes and improvements to the internal \texttt{ltxgrid} package as well.}
\end{itemize}

\subsection{REV\TeX\ ~4 Backwards Compatibility}
Documents prepared under REV\TeX\ ~4 should process correctly under REV\TeX\ ~4.1. However, the formatting of the pages and, if using Bib\TeX, the references may change.

\subsection{Submitting to APS Journals}

Authors using REV\TeX\ ~4.1 to prepare a manuscript for submission to
\textit{Physical Review Letters}, \textit{Physical Review},  \textit{Reviews of Modern Physics},
or other APS journals must also read the companion document \textit{APS Author Guide for REV\TeX\ ~4.1}
distributed with REV\TeX\ \ and follow the guidelines detailed there.

The REV\TeX\ ~4.1 distribution includes both a template
(\file{apstemplate.tex}) and a sample document (\file{apssamp.tex}).
The template is a good starting point for a manuscript. In the
following sections are instructions that should be sufficient for
creating a paper using REV\TeX\ ~4.1.

Further information about submissions to the American
Physical Society may be found at \url{http://publish.aps.org/}.

\subsection{Submitting to AIP Journals}

REV\TeX\ ~4.1 includes support for the journals of the American Institute of Physics.
The style files and authoring guides for these journals are distributed as part
REV\TeX\ ~4.1 distribution. The distribution includes both a template
(\file{aiptemplate.tex}) and a sample document (\file{aipsamp.tex}).
The template is a good starting point for a manuscript. In the
following sections are instructions that should be sufficient for
creating a paper using REV\TeX\ ~4.1.

More information may be found at
\url{http://www.aip.org/pubservs/compuscript.html}. Please consult the \textit{Author's Guide to AIP Substyles for REV\TeX\ ~4.1} for more information about submissions to AIP journals, AIP styles files, and other AIP-specific information.

\subsection{Contact Information}\label{sec:aipresources}%
Any bugs, problems, or inconsistencies with REV\TeX\ \ or the APS journal style files should be reported to
REV\TeX\ \ support at \verb+revtex@aps.org+. Reports should include information on the error and a \textit{small}
sample document that manifests the problem if possible (please don't send large files!). Issues related to the AIP journal styles should be sent directly to \verb+tex@aip.org+.

\section{Some \LaTeXe\ Basics}
REV\TeX\ ~4.1 must sometimes patch the underlying
\LaTeX\ kernel. This means that REV\TeX\ ~4.1 requires a fairly recent version of
\LaTeXe. Versions prior to 2005/12/01 may not work
correctly. REV\TeX\ ~4.1 will be maintained to be compatible with future
versions of \LaTeXe.

\subsection{Useful \LaTeXe\ Markup}
\LaTeXe\ markup is the preferred way to accomplish many basic tasks.

\subsubsection{Fonts}

Because REV\TeX\ ~4.1 is based upon \LaTeXe, it inherits all of the
macros used for controlling fonts. Of particular importance are the
\LaTeXe\ macros \cmd{\textit}, \cmd{\textbf}, \cmd{\texttt} for changing to
an italic, bold, or typewriter font respectively. One should always
use these macros rather than the lower-level \TeX\ macros \cmd{\it},
\cmd{\bf}, and \cmd{\tt}. The \LaTeXe\ macros offer
improvements such as better italic correction and scaling in super-
and subscripts for example. Table~\ref{tab:fonts}
summarizes the font selection commands in \LaTeXe.

\begin{table}
\caption{\label{tab:fonts}\LaTeXe\ font commands}
\begin{ruledtabular}
\begin{tabular}{ll}
\multicolumn{2}{c}{\textbf{Text Fonts}}\\
\textbf{Font command} & \textbf{Explanation} \\
\cmd\textit\marg{text}  & Italics\\
\cmd\textbf\marg{text}  & Boldface\\
\cmd\texttt\marg{text}  & Typewriter\\
\cmd\textrm\marg{text}  & Roman\\
\cmd\textsl\marg{text}  & Slanted\\
\cmd\textsf\marg{text}  & Sans Serif\\
\cmd\textsc\marg{text}  & Small Caps\\
\cmd\textmd\marg{text}  & Medium Series\\
\cmd\textnormal\marg{text} & Normal Series\\
\cmd\textup\marg{text}  & Upright Series\\
  &\\
\multicolumn{2}{c}{\textbf{Math Fonts}}\\
\cmd\mathit\marg{text}  & Math Italics\\
\cmd\mathbf\marg{text}  & Math Boldface\\
\cmd\mathtt\marg{text}  & Math Typewriter\\
\cmd\mathsf\marg{text}  & Math Sans Serif\\
\cmd\mathcal\marg{text}  & Calligraphic\\
\cmd\mathnormal\marg{text} & Math Normal\\
\cmd\bm\marg{text}& Bold math for Greek letters\\
                  & and other symbols\\
\cmd\mathfrak\marg{text}\footnotemark[1]  & Fraktur\\
\cmd\mathbb\marg{text}\footnotemark[1] & Blackboard Bold\\
\end{tabular}
\end{ruledtabular}
\footnotetext[1]{Requires \classname{amsfonts} or \classname{amssymb} class option}
\end{table}

\subsubsection{User-defined macros}
\LaTeXe\ provides several macros that enable users to easily create new
macros for use in their manuscripts:
\begin{itemize}
\footnotesize
\item \cmd\newcommand\marg{\\command}\oarg{narg}\oarg{opt}\marg{def}
\item \cmd\newcommand\verb+*+\marg{\\command}\oarg{narg}\oarg{opt}\marg{def}
\item \cmd\renewcommand\marg{\\command}\oarg{narg}\oarg{opt}\marg{def}
\item \cmd\renewcommand\verb+*+\marg{\\command}\oarg{narg}\oarg{opt}\marg{def}
\item \cmd\providecommand\marg{\\command}\oarg{narg}\oarg{opt}\marg{def}
\item \cmd\providecommand\verb+*+\marg{\\command}\oarg{narg}\oarg{opt}\marg{def}
\end{itemize}
Here \meta{\\command} is the name of the macro being defined,
\meta{narg} is the number of arguments the macro takes,
\meta{opt} are optional default values for the arguments, and
\meta{def} is the actually macro definiton. \cmd\newcommand\ creates a
new macro, \cmd\renewcommand\ redefines a previously defined macro,
and \cmd\providecommand\ will define a macro only if it hasn't
been defined previously. The *-ed versions are an optimization that
indicates that the macro arguments will always be ``short'' arguments. This is
almost always the case, so the *-ed versions should be used whenver
possible.

The use of these macros is preferred over using plain \TeX's low-level
macros such as
\cmd\def{},\cmd\edef{}, and \cmd\gdef{}. APS authors must follow the
\textit{APS Author Guide for REV\TeX\ ~4.1} when defining macros.

\subsubsection{Symbols}

\LaTeXe\ has added some convenient commands for some special symbols
and effects. These are summarized in Table~\ref{tab:special}. See
\cite{Guide} for details.

\begin{table}
\caption{\label{tab:special}\LaTeXe\ commands for special symbols and effects}
\begin{ruledtabular}
\begin{tabular}{lc}
Command & Symbol/Effect\\
\cmd\textemdash & \textemdash\\
\cmd\textendash & \textendash\\
\cmd\textexclamdown & \textexclamdown\\
\cmd\textquestiondown & \textquestiondown\\
\cmd\textquotedblleft & \textquotedblleft\\
\cmd\textquotedblright & \textquotedblright\\
\cmd\textquoteleft & \textquoteleft\\
\cmd\textquoteright & \textquoteright\\
\cmd\textbullet   & \textbullet\\
\cmd\textperiodcentered & \textperiodcentered\\
\cmd\textvisiblespace & \textvisiblespace\\
\cmd\textcompworkmark & Break a ligature\\
\cmd\textcircled\marg{char} & Circle a character\\
\end{tabular}
\end{ruledtabular}
\end{table}

\LaTeXe\ provides additional symbols in a
separate package called \classname{latexsym}. To use these symbols, include
the package using:
\begin{verbatim}
\usepackage{latexsym}
\end{verbatim}

\subsection{Using \LaTeXe\ packages with REV\TeX\ }\label{sec:usepackage}%

Many \LaTeXe\ packages are available, for instance, on CTAN at
\url{http://www.ctan.org/tex-archive/macros/latex/required/}
and at
\url{http://www.ctan.org/tex-archive/macros/latex/contrib/}
or may be available on other distribution media, such as the \TeX\
Live CD-ROM \url{http://www.tug.org/texlive/}.  Some of these packages
are automatically loaded by REV\TeX\ ~4.1 when certain class options are
invoked and are, thus, ``required.''  They will either be distributed
with REV\TeX\ \ or are already included with a standard \LaTeXe\
distribution.

Required packages are automatically loaded by REV\TeX\ \ on an as-needed
basis.  Other packages should be loaded using the
\cmd\usepackage\ command. To load the
\classname{hyperref} package, the document preamble might look like:
\begin{verbatim}
\documentclass{revtex}
\usepackage{hyperref}
\end{verbatim}

Some common (and very useful) \LaTeXe\ packages are \textit{a priori}
important enough that REV\TeX\ ~4.1 has been designed to be specifically
compatible with them.
A bug stemming from the use of one of these packages in
conjunction with any of the APS journals may be reported by contacting
REV\TeX\ \ support.
\begin{description}
\item[\textbf{AMS packages}] REV\TeX\ ~4.1 is compatible with and depends
 upon the AMS packages
\classname{amsfonts},
\classname{amssymb}, and
\classname{amsmath}. In fact, REV\TeX\ ~4.1 requires use of these packages
to accomplish some common tasks. See Section~\ref{sec:math} for more.
REV\TeX\ ~4.1 requires version 2.0 or higher of the AMS-\LaTeX\ package.

\item[\textbf{array and dcolumn}]
The \classname{array} and \classname{dcolumn} packages are part of
\LaTeX's required suite of packages. \classname{dcolumn} is required
to align table columns on decimal points (and it in turn depends upon
the \classname{array} package).

\item[\textbf{longtable}]
\file{longtable.sty} may be used for large tables that will span more than one
page. REV\TeX\ ~4.1 dynamically applies patches to longtable.sty so that
it will work in two-column mode.

\item[\textbf{hyperref}] \file{hyperref.sty} is a package by Sebastian Rahtz that is
used for putting hypertext links into \LaTeXe\ documents.
REV\TeX\ ~4.1 has hooks to allow e-mail addresses and URL's to become
hyperlinks if \classname{hyperref} is loaded.

\item[\textbf{lineno}] REV\TeX\ ~4.1 improves compatibility with \classname{lineno.sty}. This package should only be loaded via the new \classoption{linenumbers} class option. See Section~\ref{sec:lineno} for more information.

\item[\textbf{lscape}] REV\TeX\ ~4.1 improves compatibility with \classname{lscape.sty}.

\item[\textbf{geometry}] REV\TeX\ ~4.1 improves compatibility with \classname{geometry.sty}.

\item[\textbf{colortbl}] REV\TeX\ ~4.1 improves compatibility with \classname{colortbl.sty}.

\end{description}

Other packages will conflict with REV\TeX\ ~4.1 and should be
avoided. Usually such a conflict arises because the package adds
enhancements that REV\TeX\ ~4.1 already includes. Here are some common
packages that clash with REV\TeX\ ~4.1:
\begin{description}
\item[\textbf{multicol}] \file{multicol.sty} is a package by Frank Mittelbach
that adds support for multiple columns. In fact, early versions of
REV\TeX\ ~4.1 used \file{multicol.sty} for precisely this. REV\TeX\ ~4.1
incorporates its own support for multiple-column typesetting.

\item[\textbf{cite}] Donald Arseneau's \file{cite.sty} is often used to provide
support for sorting a \cmd\cite\ command's arguments into numerical
order and to collapse consecutive runs of reference numbers. REV\TeX\ ~4.1
has this functionality built-in already via the \classname{natbib} package.

\item[\textbf{mcite}] REV\TeX\ ~4.1 already contains a lot of this
functionality through its updated syntax for the \cmd\cite\ command and
the latest  \classname{natbib} package.

\item[\textbf{endfloat}] The same functionality can be accomplished
using the \classoption{endfloats} class option.

\item[\textbf{float}] \texttt{float.sty} provides a mechanism for creating new float classes with just a few commands. REV\TeX\ ~4.1 has limited compatible with float.sty. If attempting to use this package, be sure to put any \cmd\newfloat\ commands after the \verb+\begin{document}+ line.

\end{description}

\section{The Document Preamble}

The preamble of a \LaTeX\ document is the set of commands that precede
the \envb{document} line. It contains a
\cmd\documentclass\ line to load the REV\TeX\ ~4.1 class (\textit{i.e.},
all of the REV\TeX\ ~4.1 macro definitions), \cmd\usepackage\ macros to
load other macro packages, and other macro definitions.

\subsection{The \emph{documentclass} line}
The basic formatting of the manuscript is controlled by setting
\emph{class options} using
\cmd\documentclass\oarg{options}\aarg{\classname{revtex4-1}}.
The optional arguments that appear in the square brackets control the layout of the
document. At this point, one only needs to choose:
\begin{itemize}
\item Either the \classoption{aps} (default) or \classoption{aip} society option
\item One of the chosen society's journal styles such as \classoption{prl} or \classoption{apl}
\item A layout option such as \classoption{preprint} (single-column formatting), \classoption{reprint} (an approximation
to the selected journal's actual layout which may be one- or two-column depending on the journal), or \classoption{twocolumn}
\end{itemize}
Usually, one would want to use \classoption{preprint} for draft papers. Paper size options are also
available as well. In particular, \classoption{a4paper} is available
as well as the rest of the standard \LaTeX\ paper sizes. A
full list of class options is given in the \textit{REV\TeX\ ~4.1 Command
and Options Summary}.

\subsection{Loading other packages}
Other packages may be loaded into a REV\TeX\ ~4.1 document by using the
standard \LaTeXe\ \cmd\usepackage\ command. For instance, to load
the \classoption{graphics} package, one would use
\verb+\usepackage{graphics}+.

\section{The Front Matter}\label{sec:front}

After choosing the basic look and feel of the document by selecting
the appropriate class options and loading in whatever other macros are
needed, one is ready to move on to creating a new manuscript. After
the preamble, be sure to put in a \envb{document} line (and put
in an \enve{document} as well). This section describes the macros
REV\TeX\ ~4.1 provides for formatting the front matter of the
article. The behavior and usage of these macros can be quite
different from those provided in the \LaTeXe\ \classname{article} class.
\subsection{Setting the title}

The title of the manuscript is simply specified by using the
\cmd\title\aarg{title} macro. A \verb+\\+ may be used to put a line
break in a long title.

\subsection{Specifying a date}%

The \cmd\date\marg{date} command outputs the date on the
manuscript.  Using \cmd\today\ will cause \LaTeX{} to insert the
current date whenever the file is run:
\begin{verbatim}
\date{\today}
\end{verbatim}

\subsection{Specifying authors and affiliations}

The REV\TeX\ ~4.1 macros  for specifying authors and their affiliations are designed
 to save labor for authors and during production. Authors and affiliations are
arranged into groupings called, appropriately enough, \emph{author
groups}. Each author group is a set of authors who share the same set
of affiliations. Author names are specified with the \cmd\author\
macro while affiliations (or addresses) are specified with the
\cmd\affiliation\ macro. Author groups are specified by sequences of
\cmd\author\ macros followed by \cmd\affiliation\ macros. An
\cmd\affiliation\ macro applies to all previously specified
\cmd\author\ macros which don't already have an affiliation supplied.

For example, if Bugs Bunny and Roger Rabbit are both at Looney Tune
Studios, while Mickey Mouse is at Disney World, the markup would be:
\begin{verbatim}
\author{Bugs Bunny}
\author{Roger Rabbit}
\affiliation{Looney Tune Studios}
\author{Mickey Mouse}
\affiliation{Disney World}
\end{verbatim}
The default is to display this as
\begin{center}
Bugs Bunny and Roger Rabbit\\
\emph{Looney Tune Studios}\\
Mickey Mouse\\
\emph{Disney World}\\
\end{center}
This layout style for displaying authors and their affiliations is
chosen by selecting the class option
\classoption{groupedaddress}. Journal styles usually default this option,
 so it need not be specified explicitly. The other major way of displaying this
information is to use superscripts on the authors and
affiliations. This can be accomplished by selecting the class option
\classoption{superscriptaddress}. To achieve the display
\begin{center}
Bugs Bunny,$^{1}$ Roger Rabbit,$^{1,2}$ and Mickey Mouse$^{2}$\\
\emph{$^{1}$Looney Tune Studios}\\
\emph{$^{2}$Disney World}\\
\end{center}
one would use the markup
\begin{verbatim}
\author{Bugs Bunny}
\affiliation{Looney Tune Studios}
\author{Roger Rabbit}
\affiliation{Looney Tune Studios}
\affiliation{Disney World}
\author{Mickey Mouse}
\affiliation{Disney World}
\end{verbatim}

Note that REV\TeX\ ~4.1 takes care of any commas and \emph{and}'s that join
the author names together and font selection, as well as any
superscript numbering. Only the author names and affiliations should
be given within their respective macros. See below for further information
regarding the proper way to add footnotes to author names and affiliations.

There is a third class option, \classoption{unsortedaddress}, for
controlling author/affiliation display. The default
\classoption{groupedaddress} will actually sort authors into the
approriate author groups if one chooses to specify an affiliation for
each author. The markup:
\begin{verbatim}
\author{Bugs Bunny}
\affiliation{Looney Tune Studios}
\author{Mickey Mouse}
\affiliation{Disney World}
\author{Roger Rabbit}
\affiliation{Looney Tune Studios}
\end{verbatim}
will result in the same display as for the first case given
above even though Roger Rabbit is specified after Mickey Mouse. To
avoid Roger Rabbit being moved into the same author group as Bugs
Bunny, use the
\classoption{unsortedaddress} option instead. In general, it is safest
to list authors in the order they should appear and specify
affiliations for multiple authors rather than one at a time. This will
afford the most independence for choosing the display option. Finally,
it should be mentioned that the affiliations for the
\classoption{superscriptaddress} are presented and numbered
in the order that they are encountered. These means that the order
will usually follow the order of the authors. An alternative ordering
can be forced by including a list of \cmd\affiliation\ commands before
the first \cmd{\author} in the desired order. Then use the exact same
text for each affilation when specifying them for each author.

If an author doesn't have an affiliation, the \cmd\noaffiliation\
macro may be used in the place of an \cmd\affiliation\ macro.

\subsubsection{Collaborations}

A collaboration name can be specified with the \cmd\collaboration\
command. This is very similar to the \cmd\author\ command. In REV\TeX\ ~4.1, it can
be used with both the \classoption{superscriptaddress} and \classoption{groupedaddress} class options. The
\cmd\collaboration\ command should appear at the end of the list of
authors. The collaboration name will be appear centered in parentheses
between the list of authors and the list of
affiliations. Because collaborations
don't normally have affiliations, one needs to follow the
\cmd\collaboration\ with \cmd\noaffiliation.

\subsubsection{Footnotes for authors, collaborations, affiliations or title}\label{sec:footau}

Often one wants to specify additional information associated with an
author, collaboration, or affiliation such as an e-mail address, an
alternate affiliation, or some other ancillary information.
REV\TeX\ ~4.1 introduces several new macros just for this purpose. They
are:
\begin{itemize}
\item\cmd\email\oarg{optional text}\aarg{e-mail address}
\item\cmd\homepage\oarg{optional text}\aarg{URL}
\item\cmd\altaffiliation\oarg{optional text}\aarg{affiliation}
\item\cmd\thanks\aarg{miscellaneous text}
\end{itemize}
In the first three, the \emph{optional text} will be prepended before the
actual information specified in the required argument. In the APS journal style files, \cmd\email\ and \cmd\homepage\ no longer have a default value. However, in the AIP styles, each have a default text for their optional arguments
(`Electronic address:' and `URL:' respectively). The \cmd\thanks\
macro should only be used if one of the other three do not apply. Any
author name can have multiple occurences of these four macros. Note
that unlike the
\cmd\affiliation\ macro, these macros only apply to the \cmd\author\
that directly precedes it. Any \cmd\affiliation\ \emph{must} follow
the other author-specific macros. A typical usage might be as follows:
\begin{verbatim}
\author{Bugs Bunny}
\email[E-mail me at: ]{bugs@looney.com}
\homepage[Visit: ]{http://looney.com/}
\altaffiliation[Permanent address: ]
                     {Warner Brothers}
\affiliation{Looney Tunes}
\end{verbatim}
This would result in the footnote ``E-mail me at: \texttt{bugs@looney.com},
Visit: \texttt{http://looney.com/}, Permanent address: Warner
Brothers'' being attached to Bugs Bunny. Note that:
\begin{itemize}
\item Only an e-mail address, URL, or affiliation should go in the
required argument in the curly braces.
\item The font is automatically taken care of.
\item An explicit space is needed at the end of the optional text if one is
desired in the output.
\item Use the optional arguments to provide customized
text only if there is a good reason to.
\end{itemize}

The \cmd\collaboration\ , \cmd\affiliation\ , or even \cmd\title\ can
also have footnotes attached via these commands. If any ancillary data
(\cmd\thanks, \cmd\email, \cmd\homepage, or
\cmd\altaffiliation) are given in the wrong context (e.g., before any
\cmd\title, \cmd\author, \cmd\collaboration, or \cmd\affiliation\
command has been given), then a warning is given in the \TeX\ log, and
the command is ignored.

Duplicate sets of ancillary data are merged, giving rise to a single
shared footnote. However, this only applies if the ancillary data are
identical: even the order of the commands specifying the data must be
identical. Thus, for example, two authors can share a single footnote
indicating a group e-mail address.

Duplicate \cmd\affiliation\ commands may be given in the course of the
front matter, without the danger of producing extraneous affiliations
on the title page. However, ancillary data should be specified for
only the first instance of any particular institution's
\cmd\affiliation\ command; a later instance with different ancillary
data will result in a warning in the \TeX\ log.

It is preferable to arrange authors into
sets. Within each set all the authors share the same group of
affiliations. For each author, give the \cmd\author\ (and appropriate
ancillary data), then follow this author group with the needed group
of \cmd\affiliation\ commands.

If affiliations have been listed before the first
\cmd\author\ macro to ensure a particular ordering, be sure
that any later \cmd\affiliation\ command for the given institution is
an exact copy of the first, and also ensure that no ancillary data is
given in these later instances.

Each journal class option has a default behavior for the placement of these
ancillary information footnotes. For instance, the \classoption{prb} option puts all
such footnotes at the start of the bibliography while the \classoption{prl}
journal styles displays them on the first page. One can override a
journal style's default behavior by specifying explicitly the class
option
\classoption{bibnotes} (puts the footnotes at the start of the
bibliography) or \classoption{nobibnotes} (puts them on the first page).
Please consult the documentation for the various journal style files for further information.

\subsubsection{Specifying first names and surnames}

Many authors have names in which either the surname appears first
or in which the surname is made up of more than one name. To ensure
that such names are accurately captured for indexing and other
purposes, the \cmd\surname\ macro should be used to indicate which portion
of a name is the surname. Similarly, there is a \cmd\firstname\ macro
as well, although usage of \cmd\surname\ should be sufficient. If an
author's surname is a single name and written last, it is not
necessary to use these macros. These macros do nothing but indicate
how a name should be indexed. Here are some examples:
\begin{verbatim}
\author{Andrew \surname{Lloyd Weber}}
\author{\surname{Mao} Tse-Tung}
\end{verbatim}

\subsection{The abstract}
An abstract for a paper is specified by using the \env{abstract}
environment:
\begin{verbatim}
\begin{abstract}
Text of abstract
\end{abstract}
\end{verbatim}
Note that in REV\TeX\ ~4.1 the abstract must be specified before the
\cmd\maketitle\ command and there is no need to embed it in an explicit
minipage environment.

\subsubsection{Structured abstracts}
A new feature in REV\TeX\ ~4.1 is support for \textit{structured abstracts}. A ``structured" abstract is an abstract divided into labeled sections. For instance, \textit{Physical Review C} would like authors to provide abstracts with sections summarizing the paper's  \textbf{Background}, \textbf{Purpose}, \textbf{Method}, \textbf{Results}, and \textbf{Conclusions}. This can be accomplished by using the \texttt{description} environment within the \texttt{abstract} environment.  For example:
\begin{verbatim}
\begin{abstract}
\begin{description}
\item[Background] This part would describe the
context needed to understand what the paper
is about.
\item[Purpose] This part would state the purpose
of the present paper.
\item[Method] This part describe the methods
used in the paper.
\item[Results] This part would summarize the
results.
\item[Conclusions] This part would state the
conclusions of the paper.
\end{description}
\end{abstract}
\end{verbatim}

\subsection{PACS codes}
APS and AIP authors are asked to supply suggested PACS codes with their
submissions. The \cmd\pacs\ macro is provided as a way to do this:
\begin{verbatim}
\pacs{23.23.+x, 56.65.Dy}
\end{verbatim}
The actual display of the PACS numbers below the abstract is
controlled by two class options: \classoption{showpacs} and
\classoption{noshowpacs}. In particular, this is now independent of
the \classoption{preprint} option. \classoption{showpacs} must be
explicitly included in the class options to display the PACS codes.

\subsection{Keywords}
A \cmd\keywords\ macro may also be used to indicate keywords for the
article.
\begin{verbatim}
\keywords{nuclear form; yrast level}
\end{verbatim}
This will be displayed below the abstract and PACS (if supplied). Like
PACS codes, the actual display of the the keywords is controlled by
two classoptions: \classoption{showkeys} and
\classoption{noshowkeys}. An explicit \classoption{showkeys} must be
included in the \cmd\documentclass\ line to display the keywords.

\subsection{Institutional report numbers}
Institutional report numbers can be specified using the \cmd\preprint\
macro. If the \classoption{preprintnumbers} class option is specified, these will be displayed in the upper right corner of the first page. Multiple \cmd\preprint\ macros maybe supplied (space is
limited though, so only three or less may actually fit). Please note that the \classoption{preprint} class option does not automatically invoke \classoption{preprintnumbers}.

\subsection{maketitle}
After specifying the title, authors, affiliations, abstract, PACS
codes, and report numbers, the final step for formatting the front
matter of the manuscript is to execute the \cmd\maketitle\ macro by
simply including it:
\begin{verbatim}
\maketitle
\end{verbatim}
The \cmd\maketitle\ macro must follow all of the macros listed
above. The macro will format the front matter in accordance with the various
class options that were specified in the
\cmd\documentclass\ line (either implicitly through defaults or
explicitly).

\section{The body of the paper}

For typesetting the body of a paper, REV\TeX\ ~4.1 relies heavily on
standard \LaTeXe\ and other packages (particulary those that are part
of AMS-\LaTeX). Users unfamiliar with these packages should read the
following sections carefully.

\subsection{Section headings}

Section headings are input as in \LaTeX.
The output is similar, with a few extra features.

Four levels of headings are available in REV\TeX\ {}:
\begin{quote}
\cmd\section\marg{title text}\\
\cmd\subsection\marg{title text}\\
\cmd\subsubsection\marg{title text}\\
\cmd\paragraph\marg{title text}
\end{quote}

Use the starred form of the command to suppress the automatic numbering; e.g.,
\begin{verbatim}
\section*{Introduction}
\end{verbatim}

To label a section heading for cross referencing, best practice is to
place the \cmd\label\marg{key} within the argument specifying the heading:
\begin{verbatim}
\section{\label{sec:intro}Introduction}
\end{verbatim}

In some journal substyles, such as those of the APS,
all text in the \cmd\section\ command is automatically set uppercase.
If a lowercase letter is needed, use \cmd\lowercase\aarg{x}.
For example, to use ``He'' for helium in a \cmd\section\marg{title text} command, type
\verb+H+\cmd\lowercase\aarg{e} in \marg{title text}.

Use \cmd\protect\verb+\\+ to force a line break in a section heading.
(Fragile commands must be protected in section headings, captions, and
footnotes and \verb+\\+ is a fragile command.)

\subsection{Paragraphs and General Text}

Paragraphs always end with a blank input line.  Because \TeX\
automatically calculates linebreaks and word hyphenation in a
paragraph, it is not necessary to force linebreaks or hyphenation.  Of
course, compound words should still be explicitly hyphenated, e.g.,
``author-prepared copy.''

Use directional quotes for quotation marks around quoted text
(\texttt{``xxx''}), not straight double quotes (\texttt{"xxx"}).
For opening quotes, use one or two backquotes; for closing quotes,
use one or two forward quotes (apostrophes).

\subsection{One-column vs. two-column layouts}\label{sec:widetext}

One of the hallmarks of \textit{Physical Review} and many of the AIP journals is their two-column
formatting. REV\TeX\ ~4.1 provides the \classoption{reprint} class option that provides for each
journal class option a close approximation to the journal's actual production formatting. Note that
the \classoption{reprint} option will give either one or two-column formatting as appropriate for the particular journal.
For most APS and AIP journals, the \classoption{reprint} option will take care of formatting the front matter
(including the abstract) as a single column and will typeset the body in two columns. REV\TeX\ ~4.1 has its own
built-in two-column formatting macros to provide well-balanced columns as well as reasonable control over the placement of floats in either
one- or two-column modes. When drafting papers, it is common to use a one-column format. This is best achieved by using the
\classoption{preprint} class option. Authors may override a particular journal's formatting by using the lower level options \classoption{onecolumn} and \classoption{twocolum}, but best practice is to stick with the \classoption{preprint} and \classoption{reprint} options.

Please note that the \classoption{reprint} class option is only an \textit{approximation} of a journal's final layout. Because of font differences, figure rescaling, and other factors, authors should not expect the \classoption{reprint} option to give fully accurate estimates of an article's ultimate length after being typeset for the journal.

Occasionally it is necessary to change the formatting from two-column to
one-column to better accommodate very long equations that are more
easily read when typeset to the full width of the page. This is
accomplished using the \env{widetext} environment:
\begin{verbatim}
\begin{widetext}
long equation goes here
\end{widetext}
\end{verbatim}
In two-column mode, this will temporarily return to one-column mode,
balancing the text before the environment into two short columns, and
returning to two-column mode after the environment has
finished. REV\TeX\ ~4.1 will also add horizontal rules to guide the
reader's eye through what may otherwise be a confusing break in the
flow of text. The
\env{widetext} environment has no effect on the output under the
\classoption{preprint} class option because this already uses
one-column formatting.

Use of the \env{widetext} environment should be restricted to the bare
minimum of text that needs to be typeset this way. However, short pieces
of paragraph text and/or math between nearly contiguous wide equations
should be incorporated into the surrounding wide sections.

Low-level control over the column grid can be accomplished with the
\cmd\onecolumngrid\ and \cmd\twocolumngrid\ commands. Using these, one
can avoid the horizontal rules added by \env{widetext}. These commands
should only be used if absolutely necessary. Wide figures and tables
should be accommodated using the proper \verb+*+ environments.

\subsection{Cross-referencing}\label{sec:xrefs}

REV\TeX\ {} inherits the \LaTeXe\ features for labeling and cross-referencing
section headings, equations, tables, and figures. This section
contains a simplified explanation of these cross-referencing features.
The proper usage in the context of section headings, equations,
tables, and figures is discussed in the appropriate sections.

Cross-referencing depends upon the use of ``tags,'' which are defined by
the user.  The \cmd\label\marg{key} command is used to identify tags for
REV\TeX\ . Tags are strings of characters that serve to label section
headings, equations, tables, and  figures that replace explicit,
by-hand numbering.

Files that use cross-referencing (and almost all manuscripts do)
need to be processed through REV\TeX\ \ at least twice to
ensure that the tags have been properly linked to appropriate numbers.
If any tags are added in subsequent editing sessions,
\LaTeX{} will display a warning message in the log file that ends with
\texttt{... Rerun to get cross-references right}.
Running the file through REV\TeX\ \ again (possibly more than once) will
resolve the cross-references.  If the error message persists, check
the labels; the same \marg{key} may have been used to label more than one
object.

Another \LaTeX\ warning is \texttt{There were undefined references},
which indicates the use of a key in a \cmd\ref\ without ever
using it in a \cmd\label\ statement.

REV\TeX\ {} performs autonumbering exactly as in standard \LaTeX.
When the file is processed for the first time,
\LaTeX\ creates an auxiliary file (with the \file{.aux} extension) that
records the value of each \meta{key}.  Each subsequent run retrieves
the proper number from the auxiliary file and updates the auxiliary
file.  At the end of each run, any change in the value of a \meta{key}
produces a \LaTeX\ warning message.

Note that with footnotes appearing in the bibliography, extra passes
of \LaTeX\ may be needed to resolve all cross-references. For
instance, putting a \cmd\cite\ inside a \cmd\footnote\ will require at
least three passes.

Using the \classname{hyperref} package to create hyperlinked PDF files
will cause reference ranges to be expanded to list every
reference in the range. This behavior can be avoided by using the
\classname{hypernat} package available from \url{www.ctan.org}.

\subsection{Acknowledgments}
Use the \env{acknowledgments} environment for an acknowledgments
section.  Depending on the journal substyle, this element may be
formatted as an unnumbered section title \textit{Acknowledgments} or
simply as a paragraph. Please note the spelling of
``acknowledgments.''
\begin{verbatim}
\begin{acknowledgments}
The authors would like to thank...
\end{acknowledgments}
\end{verbatim}

\subsection{Appendices}
The \cmd
\section{<heading>}, \subsection{<heading>}+,
\verb+\subsubsection{<heading>}+ & Start a new section or subsection.\\
\verb+\section*{<heading>}+ & Start a new section without a number.\\
\verb+
\section{New Syntax and Features in REV\TeX\ ~4.1}
REV\TeX\ ~4.1 introduces support for more journals, several new commands, and some new syntax. This section outlines these changes. \textbf{A document using these new features will not process under REV\TeX\ ~4}. See Sec.~\ref{sec:additional} for more details about these items.

\begin{itemize}
\item \textbf{Added support for APS journal \textit{Physical Review Special Topics -- Physics Education Research}}.
\item \textbf{Added support for AIP journals.} There is now an explicit \texttt{aip} society option along with support for AIP journals. Please see the \textit{Author's Guide to AIP Substyles for REV\TeX\ ~4.1}. In addition, REV\TeX\ ~4.1 provides an extensible system for the easy addition of new collections of journals.
\item \textbf{Endnotes now ordered correctly.} Endnotes in the bibliography now appear in the correct order, interleaved with citations.
\item \textbf{Multiple references in a single citation supported using a special starred (*) argument to the \cmd\cite\ command.} One of the major new features in 4.1 made possible by the joint work on \texttt{natbib 8.3}. Multiple Bib\TeX\ entries can be combined into a single \cmd\bibitem\ command.
\item \textbf{Bib\TeX\ style files can now display journal article titles in the bibliography.} Use the \texttt{longbibliography} class option.
\item \textbf{Free form text can be prepended and appended to a bibliographic entry using the special starred (*) argument to the \cmd\cite\ command.} Often a citation in the bibliography will have explanatory text such as \textit{See also} or \textit{and references therein} before and after the actual citation. The new REV\TeX\ ~4.1 \cmd\cite\ command allows the specification of both text to precede and follow a citation.
\item \textbf{Structured Abstracts.} Use of the \texttt{description} environment in abstracts now provides for ``structured" abstracts.
\item \textbf{Figures referring to videos now supported.} A ``figure" may now be labeled as a \textbf{Video} by using the \texttt{video} environment. A frame from the video may be included in the figure and a URL to link the caption's label to the online video also may be included. There is also a \cmd\listofvideos\ command.
\item \textbf{Better support for arXiv.org in Bib\TeX\ } Three more Bib\TeX\ fields have been added: \texttt{SLACcitation}, \texttt{archivePrefix}, and \texttt{primaryClass} in addition to the existing field \texttt{eprint}.
\end{itemize}

\section{Bug Fixes in REV\TeX\ ~4.1}
One of the main goals of REV\TeX\ ~4.1 is, of course, to fix the bugs that were released in REV\TeX\ ~4. The following is a list of bugs that have been fixed.

\begin{itemize}
\item \textbf{Improved Bib\TeX\ \texttt{bst} files.} In addition to the new features above, numerous other improvements to the APS \texttt{bst} files have been made, including support for displaying journal article titles and many fixes for \textit{Reviews of Modern Physics}. Also, long author lists are no longer automatically truncated.
\item \textbf{\cmd\footnote\ in \cmd\widetext\ and \texttt{table*} environments improved.} \cmd\footnote\relax s in  the \cmd\widetext\ or \texttt{table*} environments are now correctly placed and formatted.
\item \textbf{Email addresses no longer print twice on papers less than one page long.}
\item \textbf{\texttt{eqnarray} alignment improved.}
\item \textbf{\cmd\collaboration\ can be used with the \texttt{groupedaddress} option now.}
\item \textbf{\texttt{letterpaper} now ensured as default paper size.}
\item \textbf{Table of Contents formatting improved.}
\item \textbf{Support for \texttt{longtable} and \texttt{lscape} packages improved.}
\item \textbf{\texttt{reftest} restored.}
\item \textbf{Compatibility with the \texttt{geometry, lineno,} and \texttt{colortbl} packages improved.} For line numbering, rather than using \texttt{lineno.sty} directly, the \texttt{linenumbers} class option should be used (this will call in \texttt{lineno.sty} with a proper set of default parameters).
\item \texttt{hyperref} \textbf{fixes}. Improved compatibility between footnotes and the \texttt{hyperref} package. In particular, table footnotes were fixed. More anchors for \texttt{hyperref} were also added (titlepage, abstract, and acknowledgements).
\item \textbf{Documents can have more than 256 \cmd\cite\ commands now.}
\item \textbf{\cmd\listoffigures\ and \cmd\listoftables\ fixed.}
\item \textbf{Figure and table labels in captions now reflect proper APS style.}
\item \textbf{RMP style files conform better to RMP style guidelines.}
\item \textbf{Section heading upper-casing improved.}
\item \textbf{Repeated characters at start of affiliation no longer disappear when using \texttt{groupedaddress} option.}
\item \textbf{There have been many other bug fixes and improvements to the internal \texttt{ltxgrid} package as well.}
\end{itemize}

\section{REV\TeX\ ~4 Backwards Compatibility}
The vast majority of documents prepared under REV\TeX\ ~4 should process correctly under REV\TeX\ ~4.1. However, the formatting of the pages and, if using Bib\TeX, the references may change. Also, the behavior of some macros may be different. For instance, the \texttt{title} macro now requires the use of \texttt{protect} for fragile arguments. This may cause some documents prepared under REV\TeX\ ~4 to fail under 4.1. Some macro packages that depend on the internals of REV\TeX\ ~4 may also no longer work.  Documents using those packages will, of course, also will not process under 4.1.

\section{\label{sec:additional}Additional Details}

\subsection{Multiple references in a single bibliography entry}
One of the most frequently requested features since the release of REV\TeX\ ~4 has been to allow more than one reference to appear in a single bibliography entry when using Bib\TeX. This can now be done in REV\TeX\ ~4.1 by using a starred (*) argument to the \cmd\cite\ command. This requires the latest version of \texttt{natbib}, developed in conjunction with REV\TeX\ ~4.1, and the new \texttt{bst} files that come with REV\TeX\ ~4.1. To combine multiple references into a single \cmd\bibitem, precede the second, third, etc. citation keys in the \cmd\cite\ command with an asterisk (*). For example \verb+\cite{bethe, *feynman, *bohr}+ will combine the \cmd\bibitem\relax s with keys \texttt{bethe}, \texttt{feynman}, and \texttt{bohr} into a single entry in the bibliography separated by semicolons.

\subsection{Prepending and/or appending text to a citation}
The expanded syntax for the  \cmd\cite\ command argument  can also be used to specify text before and/or after a citation. For instance, a citation such as:
\begin{verbatim}
[19] A similar expression was derived in
A. V. Andreev, Phys. Rev. Lett. 99, 247204
(2007) in the context of carbon nanotube
p-n junctions. The only difference is that no
integration over ky is present there.
\end{verbatim}
may be created by the following \cmd\cite\ command:
\begin{verbatim}
\cite{*[{A similar expression was derived
in }] [{ in the context of carbon nanotube
p-n junctions. The only difference is that no
integration over ky is present there.}]andreev2007]
\end{verbatim}
Please note the use of curly braces to enclose the text within the square brackets.
\subsection{Structured Abstracts}
A ``structured" abstract is an abstract divided into labeled sections. For instance, \textit{Physical Review C} would like authors to provide abstracts with sections summarizing the paper's  \textbf{Background}, \textbf{Purpose}, \textbf{Method}, \textbf{Results}, and \textbf{Conclusions}. This can be accomplished by using the \texttt{description} environment within the \texttt{abstract} environment.  For example:
\begin{verbatim}
\begin{abstract}
\begin{description}
\item[Background] This part would describe the
context needed to understand what the paper
is about.
\item[Purpose] This part would state the purpose
of the present paper.
\item[Method] This part describe the methods
used in the paper.
\item[Results] This part would summarize the
results.
\item[Conclusions] This part would state the
conclusions of the paper.
\end{description}
\end{abstract}
\end{verbatim}

\subsection{Video Environment}
Papers often refer to multimedia videos. The \texttt{video} environment is identical to the \texttt{figure} environment, but the caption will be labeled as a \textbf{Video} (with its own counter independent of figures). A URL can also be specified so that the caption label can be linked to the online video (if using the \texttt{hyperref} package). The included graphic (using \cmd\includegraphics\ from the \texttt{graphics} or \texttt{graphicx} package) would be a representation frame from the video. A \texttt{\cmd\listofvideos} is also provided.  For example:
\begin{verbatim}
\begin{video}
\includegraphics{videoframe.jpg}
\setfloatlink{http://some.video.com/fun.mov}
\caption{\label{vid:interest}This is a video of
something fun.}
\end{video}
\end{verbatim}

\subsection{Better arXiv.org support in Bib\TeX\ }

There have been substantial improvements in the REV\TeX\ \ Bib\TeX\ style files. For instance, the \texttt{.bib} entry
\begin{verbatim}
@Unpublished{Ginsparg:1988ui,
     author    = "Ginsparg, Paul H.",
     title     = "{APPLIED CONFORMAL FIELD THEORY}",
     year      = "1988",
     eprint    = "hep-th/9108028",
     archivePrefix = "arXiv",
     SLACcitation  =
}
\end{verbatim}
will include the arXiv.org e-print identifier as \texttt{arXiv:hep-th/9108028} and hyperlink it (if using \texttt{hyperref}). The newer format for arXiv identifiers with primary classificiations will produce output such as \texttt{arXiv:0905.1949 [hep-ph]}.

\begin{acknowledgments}
The development of  REV\TeX\ ~4.1 was managed by Mark Doyle (APS). The development of the new AIP style files and their accompanying documentation was managed by Susan Joy (AIP) with the help of Chris McMahon (AIP) and Rich O'Keeffe (AIP). Testing and  evaluation were done by Michele Hake (APS), Liz Belmont (AIP), Brian Goss (AIP), Alison Waldron (AIP),  and Phil Robertson (AIP). Additional detailed testing and feedback were provided by Lev Bishop (Yale).
\end{acknowledgments}

\end{document}

\section{\label{sec:level1}First-level heading:\protect\\ The line
break was forced \lowercase{via} \textbackslash\textbackslash}

This sample document demonstrates proper use of REV\TeX~4.1 (and
\LaTeXe) in manuscripts prepared for submission to AIP
journals. Further information can be found in the documentation included in the distribution or available at
\url{http://authors.aip.org} and in the documentation for
REV\TeX~4.1 itself.

When commands are referred to in this example file, they are always
shown with their required arguments, using normal \TeX{} format. In
this format, \verb+#1+, \verb+#2+, etc. stand for required
author-supplied arguments to commands. For example, in
\verb+\section{#1}+ the \verb+#1+ stands for the title text of the
author's section heading, and in \verb+\title{#1}+ the \verb+#1+
stands for the title text of the paper.

Line breaks in section headings at all levels can be introduced using
\textbackslash\textbackslash. A blank input line tells \TeX\ that the
paragraph has ended.

\subsection{\label{sec:level2}Second-level heading: Formatting}

This file may be formatted in both the \texttt{preprint} (the default) and
\texttt{reprint} styles; the latter format may be used to
mimic final journal output. Either format may be used for submission
purposes; however, for peer review and production, AIP will format the
article using the \texttt{preprint} class option. Hence, it is
essential that authors check that their manuscripts format acceptably
under \texttt{preprint}. Manuscripts submitted to AIP that do not
format correctly under the \texttt{preprint} option may be delayed in
both the editorial and production processes.

The \texttt{widetext} environment will make the text the width of the
full page, as on page~\pageref{eq:wideeq}. (Note the use the
\verb+\pageref{#1}+ to get the page number right automatically.) The
width-changing commands only take effect in \texttt{twocolumn}
formatting. It has no effect if \texttt{preprint} formatting is chosen
instead.

\subsubsection{\label{sec:level3}Third-level heading: Citations and Footnotes}

Citations in text refer to entries in the Bibliography;
they use the commands \verb+\cite{#1}+ or \verb+\onlinecite{#1}+.
Because REV\TeX\ uses the \verb+natbib+ package of Patrick Daly,
its entire repertoire of commands are available in your document;
see the \verb+natbib+ documentation for further details.
The argument of \verb+\cite+ is a comma-separated list of \emph{keys};
a key may consist of letters and numerals.

By default, citations are numerical; \cite{feyn54} author-year citations are an option.
To give a textual citation, use \verb+\onlinecite{#1}+: (Refs.~\onlinecite{witten2001,epr,Bire82}).
REV\TeX\ ``collapses'' lists of consecutive numerical citations when appropriate.
REV\TeX\ provides the ability to properly punctuate textual citations in author-year style;
this facility works correctly with numerical citations only with \texttt{natbib}'s compress option turned off.
To illustrate, we cite several together \cite{feyn54,witten2001,epr,Berman1983},
and once again (Refs.~\onlinecite{epr,feyn54,Bire82,Berman1983}).
Note that, when numerical citations are used, the references were sorted into the same order they appear in the bibliography.

A reference within the bibliography is specified with a \verb+\bibitem{#1}+ command,
where the argument is the citation key mentioned above.
\verb+\bibitem{#1}+ commands may be crafted by hand or, preferably,
generated by using Bib\TeX.
The AIP styles for REV\TeX~4 include Bib\TeX\ style files
\verb+aipnum.bst+ and \verb+aipauth.bst+, appropriate for
numbered and author-year bibliographies,
respectively.
REV\TeX~4 will automatically choose the style appropriate for
the document's selected class options: the default is numerical, and
you obtain the author-year style by specifying a class option of \verb+author-year+.

This sample file demonstrates a simple use of Bib\TeX\
via a \verb+\bibliography+ command referencing the \verb+aipsamp.bib+ file.
Running Bib\TeX\ (in this case \texttt{bibtex
aipsamp}) after the first pass of \LaTeX\ produces the file
\verb+aipsamp.bbl+ which contains the automatically formatted
\verb+\bibitem+ commands (including extra markup information via
\verb+\bibinfo+ commands). If not using Bib\TeX, the
\verb+thebibiliography+ environment should be used instead.

\paragraph{Fourth-level heading is run in.
Footnotes are produced using the \verb+\footnote{#1}+ command.
Numerical style citations put footnotes into the
bibliography\footnote{Automatically placing footnotes into the bibliography requires using BibTeX to compile the bibliography.}.
Author-year and numerical author-year citation styles (each for its own reason) cannot use this method.
Note: due to the method used to place footnotes in the bibliography, \emph{you
must re-run BibTeX every time you change any of your document's
footnotes}.

\section{Math and Equations}
Inline math may be typeset using the \verb+$+ delimiters. Bold math
symbols may be achieved using the \verb+bm+ package and the
\verb+\bm{#1}+ command it supplies. For instance, a bold $\alpha$ can
be typeset as \verb+$\bm{\alpha}$+ giving $\bm{\alpha}$. Fraktur and
Blackboard (or open face or double struck) characters should be
typeset using the \verb+\mathfrak{#1}+ and \verb+\mathbb{#1}+ commands
respectively. Both are supplied by the \texttt{amssymb} package. For
example, \verb+$\mathbb{R}$+ gives $\mathbb{R}$ and
\verb+$\mathfrak{G}$+ gives $\mathfrak{G}$

In \LaTeX\ there are many different ways to display equations, and a
few preferred ways are noted below. Displayed math will center by
default. Use the class option \verb+fleqn+ to flush equations left.

Below we have numbered single-line equations, the most common kind:
\begin{eqnarray}
\chi_+(p)\alt{\bf [}2|{\bf p}|(|{\bf p}|+p_z){\bf ]}^{-1/2}
\left(
\begin{array}{c}
|{\bf p}|+p_z\\
px+ip_y
\end{array}\right)\;,
\\
\left\
 \openone234567890abc123\alpha\beta\gamma\delta1234556\alpha\beta
 \frac{1\sum^{a}_{b}}{A^2
\right\
\label{eq:one}.
\end{eqnarray}
Note the open one in Eq.~(\ref{eq:one}).

Not all numbered equations will fit within a narrow column this
way. The equation number will move down automatically if it cannot fit
on the same line with a one-line equation:
\begin{equation}
\left\{
 ab12345678abc123456abcdef\alpha\beta\gamma\delta1234556\alpha\beta
 \frac{1\sum^{a}_{b}}{A^2
\right\}.
\end{equation}

When the \verb+\label{#1}+ command is used [cf. input for
Eq.~(\ref{eq:one})], the equation can be referred to in text without
knowing the equation number that \TeX\ will assign to it. Just
use \verb+\ref{#1}+, where \verb+#1+ is the same name that used in
the \verb+\label{#1}+ command.

Unnumbered single-line equations can be typeset
using the \verb+\[+, \verb+\]+ format:
\[g^+g^+ \rightarrow g^+g^+g^+g^+ \dots ~,~~q^+q^+\rightarrow
q^+g^+g^+ \dots ~. \]

\subsection{Multiline equations}

Multiline equations are obtained by using the \verb+eqnarray+
environment.  Use the \verb+\nonumber+ command at the end of each line
to avoid assigning a number:
\begin{eqnarray}
{\cal M}=&&ig_Z^2(4E_1E_2)^{1/2}(l_i^2)^{-1}
\delta_{\sigma_1,-\sigma_2}
(g_{\sigma_2}^e)^2\chi_{-\sigma_2}(p_2)\nonumber\\
&&\times
[\epsilon_jl_i\epsilon_i]_{\sigma_1}\chi_{\sigma_1}(p_1),
\end{eqnarray}
\begin{eqnarray}
\sum \vert M^{\text{viol}}_g \vert ^2&=&g^{2n-4}_S(Q^2)~N^{n-2}
        (N^2-1)\nonumber \\
 & &\times \left( \sum_{i<j}\right)
  \sum_{\text{perm}}
 \frac{1}{S_{12}}
 \frac{1}{S_{12}}
 \sum_\tau c^f_\tau~.
\end{eqnarray}
\textbf{Note:} Do not use \verb+\label{#1}+ on a line of a multiline
equation if \verb+\nonumber+ is also used on that line. Incorrect
cross-referencing will result. Notice the use \verb+\text{#1}+ for
using a Roman font within a math environment.

To set a multiline equation without \emph{any} equation
numbers, use the \verb+\begin{eqnarray*}+,
\verb+\end{eqnarray*}+ format:
\begin{eqnarray*}
\sum \vert M^{\text{viol}}_g \vert ^2&=&g^{2n-4}_S(Q^2)~N^{n-2}
        (N^2-1)\\
 & &\times \left( \sum_{i<j}\right)
 \left(
  \sum_{\text{perm}}\frac{1}{S_{12}S_{23}S_{n1}}
 \right)
 \frac{1}{S_{12}}~.
\end{eqnarray*}
To obtain numbers not normally produced by the automatic numbering,
use the \verb+\tag{#1}+ command, where \verb+#1+ is the desired
equation number. For example, to get an equation number of
(\ref{eq:mynum}),
\begin{equation}
g^+g^+ \rightarrow g^+g^+g^+g^+ \dots ~,~~q^+q^+\rightarrow
q^+g^+g^+ \dots ~. \tag{2.6$'$}\label{eq:mynum}
\end{equation}

A few notes on \verb=\tag{#1}=. \verb+\tag{#1}+ requires
\texttt{amsmath}. The \verb+\tag{#1}+ must come before the
\verb+\label{#1}+, if any. The numbering set with \verb+\tag{#1}+ is
\textit{transparent} to the automatic numbering in REV\TeX{};
therefore, the number must be known ahead of time, and it must be
manually adjusted if other equations are added. \verb+\tag{#1}+ works
with both single-line and multiline equations. \verb+\tag{#1}+ should
only be used in exceptional case - do not use it to number all
equations in a paper.

Enclosing single-line and multiline equations in
\verb+\begin{subequations}+ and \verb+\end{subequations}+ will produce
a set of equations that are ``numbered'' with letters, as shown in
Eqs.~(\ref{subeq:1}) and (\ref{subeq:2}) below:
\begin{subequations}
\label{eq:whole}
\begin{equation}
\left\{
 abc123456abcdef\alpha\beta\gamma\delta1234556\alpha\beta
 \frac{1\sum^{a}_{b}}{A^2}
\right\},\label{subeq:1}
\end{equation}
\begin{eqnarray}
{\cal M}=&&ig_Z^2(4E_1E_2)^{1/2}(l_i^2)^{-1}
(g_{\sigma_2}^e)^2\chi_{-\sigma_2}(p_2)\nonumber\\
&&\times
[\epsilon_i]_{\sigma_1}\chi_{\sigma_1}(p_1).\label{subeq:2}
\end{eqnarray}
\end{subequations}
Putting a \verb+\label{#1}+ command right after the
\verb+\begin{subequations}+, allows one to
reference all the equations in a subequations environment. For
example, the equations in the preceding subequations environment were
Eqs.~(\ref{eq:whole}).

\subsubsection{Wide equations}
The equation that follows is set in a wide format, i.e., it spans
across the full page. The wide format is reserved for long equations
that cannot be easily broken into four lines or less:
\begin{widetext}
\begin{equation}
{\cal R}^{(\text{d})}=
 g_{\sigma_2}^e
 \left(
   \frac{[\Gamma^Z(3,21)]_{\sigma_1}}{Q_{12}^2-M_W^2}
  +\frac{[\Gamma^Z(13,2)]_{\sigma_1}}{Q_{13}^2-M_W^2}
 \right)
 + x_WQ_e
 \left(
   \frac{[\Gamma^\gamma(3,21)]_{\sigma_1}}{Q_{12}^2-M_W^2}
  +\frac{[\Gamma^\gamma(13,2)]_{\sigma_1}}{Q_{13}^2-M_W^2}
 \right)\;. \label{eq:wideeq}
\end{equation}
\end{widetext}
This is typed to show the output is in wide format.
(Since there is no input line between \verb+\equation+ and
this paragraph, there is no paragraph indent for this paragraph.)
\section{Cross-referencing}
REV\TeX{} will automatically number sections, equations, figure
captions, and tables. In order to reference them in text, use the
\verb+\label{#1}+ and \verb+\ref{#1}+ commands. To reference a
particular page, use the \verb+\pageref{#1}+ command.

The \verb+\label{#1}+ should appear in a section heading, within an
equation, or in a table or figure caption. The \verb+\ref{#1}+ command
is used in the text where the citation is to be displayed.  Some
examples: Section~\ref{sec:level1} on page~\pageref{sec:level1},
Table~\ref{tab:table1}
\begin{table}
\caption{\label{tab:table1}This is a narrow table which fits into a
text column when using \texttt{twocolumn} formatting. Note that
REV\TeX~4 adjusts the intercolumn spacing so that the table fills the
entire width of the column. Table captions are numbered
automatically. This table illustrates left-aligned, centered, and
right-aligned columns.  }
\begin{ruledtabular}
\begin{tabular}{lcr}
Left\footnote{Note a.}&Centered\footnote{Note b.}&Right\\
\hline
1 & 2 & 3\\
10 & 20 & 30\\
100 & 200 & 300\\
\end{tabular}
\end{ruledtabular}
\end{table}
and Fig.~\ref{fig:epsart}.

\section{Figures and Tables}
Figures and tables are typically ``floats''; \LaTeX\ determines their
final position via placement rules.
\LaTeX\ isn't always successful in automatically placing floats where you wish them.

Figures are marked up with the \texttt{figure} environment, the content of which
imports the image (\verb+\includegraphics+) followed by the figure caption (\verb+\caption+).
The argument of the latter command should itself contain a \verb+\label+ command if you
wish to refer to your figure with \verb+\ref+.

Import your image using either the \texttt{graphics} or
\texttt{graphix} packages. These packages both define the
\verb+\includegraphics{#1}+ command, but they differ in the optional
arguments for specifying the orientation, scaling, and translation of the figure.
Fig.~\ref{fig:epsart
\begin{figure}
\includegraphics{fig_1
\caption{\label{fig:epsart} A figure caption. The figure captions are
automatically numbered.}
\end{figure}
is small enough to fit in a single column, while
Fig.~\ref{fig:wide
\begin{figure*}
\includegraphics{fig_2
\caption{\label{fig:wide}Use the \texttt{figure*} environment to get a wide
figure, spanning the page in \texttt{twocolumn} formatting.}
\end{figure*}
is too wide for a single column,
so instead the \texttt{figure*} environment has been used.

The analog of the \texttt{figure} environment is \texttt{table}, which uses
the same \verb+\caption+ command.
However, you should type your caption command first within the \texttt{table},
instead of last as you did for \texttt{figure}.

The heart of any table is the \texttt{tabular} environment,
which represents the table content as a (vertical) sequence of table rows,
each containing a (horizontal) sequence of table cells.
Cells are separated by the \verb+&+ character;
the row terminates with \verb+\\+.
The required argument for the \texttt{tabular} environment
specifies how data are displayed in each of the columns.
For instance, a column
may be centered (\verb+c+), left-justified (\verb+l+), right-justified (\verb+r+),
or aligned on a decimal point (\verb+d+).
(Table~\ref{tab:table4
\begin{table}
\caption{\label{tab:table4}Numbers in columns Three--Five have been
aligned by using the ``d'' column specifier (requires the
\texttt{dcolumn} package).
Non-numeric entries (those entries without
a ``.'') in a ``d'' column are aligned on the decimal point.
Use the
``D'' specifier for more complex layouts. }
\begin{ruledtabular}
\begin{tabular}{ccddd}
One&Two&\mbox{Three}&\mbox{Four}&\mbox{Five}\\
\hline
one&two&\mbox{three}&\mbox{four}&\mbox{five}\\
He&2& 2.77234 & 45672. & 0.69 \\
C\footnote{Some tables require footnotes.}
  &C\footnote{Some tables need more than one footnote.}
  & 12537.64 & 37.66345 & 86.37 \\
\end{tabular}
\end{ruledtabular}
\end{table}
illustrates the use of decimal column alignment.)

Extra column-spacing may be be specified as well, although
REV\TeX~4 sets this spacing so that the columns fill the width of the
table.
Horizontal rules are typeset using the \verb+\hline+
command.
The doubled (or Scotch) rules that appear at the top and
bottom of a table can be achieved by enclosing the \texttt{tabular}
environment within a \texttt{ruledtabular} environment.
Rows whose columns span multiple columns can be typeset using \LaTeX's
\verb+\multicolumn{#1}{#2}{#3}+ command
(for example, see the first row of Table~\ref{tab:table3})
\begin{table*}
\caption{\label{tab:table3}This is a wide table that spans the page
width in \texttt{twocolumn} mode. It is formatted using the
\texttt{table*} environment. It also demonstrates the use of
\textbackslash\texttt{multicolumn} in rows with entries that span
more than one column.}
\begin{ruledtabular}
\begin{tabular}{ccccc}
 &\multicolumn{2}{c}{$D_{4h}^1$}&\multicolumn{2}{c}{$D_{4h}^5$}\\
 Ion&1st alternative&2nd alternative&lst alternative
&2nd alternative\\ \hline
 K&$(2e)+(2f)$&$(4i)$ &$(2c)+(2d)$&$(4f)$ \\
 Mn&$(2g)$\footnote{The $z$ parameter of these positions is $z\sim\frac{1}{4}$.}
 &$(a)+(b)+(c)+(d)$&$(4e)$&$(2a)+(2b)$\\
 Cl&$(a)+(b)+(c)+(d)$&$(2g)$\footnote{This is a footnote in a table that spans the full page
width in \texttt{twocolumn} mode. It is supposed to set on the full width of the page, just as the caption does. }
 &$(4e)^{\text{a}}$\\
 He&$(8r)^{\text{a}}$&$(4j)^{\text{a}}$&$(4g)^{\text{a}}$\\
 Ag& &$(4k)^{\text{a}}$& &$(4h)^{\text{a}}$\\
\end{tabular}
\end{ruledtabular}
\end{table*}

The tables in this document illustrate various effects.
Tables that fit in a narrow column are contained in a \texttt{table}
environment.
Table~\ref{tab:table3} is a wide table, therefore set with the
\texttt{table*} environment.
Lengthy tables may need to break across pages.
A simple way to allow this is to specify
the \verb+[H]+ float placement on the \texttt{table} or
\texttt{table*} environment.
Alternatively, using the standard \LaTeXe\ package \texttt{longtable}
gives more control over how tables break and allows headers and footers
to be specified for each page of the table.
An example of the use of \texttt{longtable} can be found
in the file \texttt{summary.tex} that is included with the REV\TeX~4
distribution.

There are two methods for setting footnotes within a table (these
footnotes will be displayed directly below the table rather than at
the bottom of the page or in the bibliography).
The easiest
and preferred method is just to use the \verb+\footnote{#1}+
command. This will automatically enumerate the footnotes with
lowercase roman letters.
However, it is sometimes necessary to have
multiple entries in the table share the same footnote.
In this case,
create the footnotes using
\verb+\footnotemark[#1]+ and \verb+\footnotetext[#1]{#2}+.
\texttt{\#1} is a numeric value.
Each time the same value for \texttt{\#1} is used,
the same mark is produced in the table.
The \verb+\footnotetext[#1]{#2}+ commands are placed after the \texttt{tabular}
environment.
Examine the \LaTeX\ source and output for Tables~\ref{tab:table1} and
\ref{tab:table2
\begin{table}
\caption{\label{tab:table2}A table with more columns still fits
properly in a column. Note that several entries share the same
footnote. Inspect the \LaTeX\ input for this table to see
exactly how it is done.}
\begin{ruledtabular}
\begin{tabular}{cccccccc}
 &$r_c$ (\AA)&$r_0$ (\AA)&$\kappa r_0$&
 &$r_c$ (\AA) &$r_0$ (\AA)&$\kappa r_0$\\
\hline
Cu& 0.800 & 14.10 & 2.550 &Sn\footnotemark[1]
& 0.680 & 1.870 & 3.700 \\
Ag& 0.990 & 15.90 & 2.710 &Pb\footnotemark[2]
& 0.450 & 1.930 & 3.760 \\
Au& 1.150 & 15.90 & 2.710 &Ca\footnotemark[3]
& 0.750 & 2.170 & 3.560 \\
Mg& 0.490 & 17.60 & 3.200 &Sr\footnotemark[4]
& 0.900 & 2.370 & 3.720 \\
Zn& 0.300 & 15.20 & 2.970 &Li\footnotemark[2]
& 0.380 & 1.730 & 2.830 \\
Cd& 0.530 & 17.10 & 3.160 &Na\footnotemark[5]
& 0.760 & 2.110 & 3.120 \\
Hg& 0.550 & 17.80 & 3.220 &K\footnotemark[5]
&  1.120 & 2.620 & 3.480 \\
Al& 0.230 & 15.80 & 3.240 &Rb\footnotemark[3]
& 1.330 & 2.800 & 3.590 \\
Ga& 0.310 & 16.70 & 3.330 &Cs\footnotemark[4]
& 1.420 & 3.030 & 3.740 \\
In& 0.460 & 18.40 & 3.500 &Ba\footnotemark[5]
& 0.960 & 2.460 & 3.780 \\
Tl& 0.480 & 18.90 & 3.550 & & & & \\
\end{tabular}
\end{ruledtabular}
\footnotetext[1]{Here's the first, from Ref.~\onlinecite{feyn54}.}
\footnotetext[2]{Here's the second.}
\footnotetext[3]{Here's the third.}
\footnotetext[4]{Here's the fourth.}
\footnotetext[5]{And etc.}
\end{table}
for an illustration.

All AIP journals require that the initial citation of
figures or tables be in numerical order.
\LaTeX's automatic numbering of floats is your friend here:
just put each \texttt{figure} environment immediately following
its first reference (\verb+\ref+), as we have done in this example file.

\begin{acknowledgments}
We wish to acknowledge the support of the author community in using
REV\TeX{}, offering suggestions and encouragement, testing new versions,
\dots.
\end{acknowledgments}

\section{}
\subsection{}
\subsubsection{}

\section{\label{sec:level1}First-level heading:\protect\\ The line
break was forced \lowercase{via} \textbackslash\textbackslash}

This sample document demonstrates proper use of REV\TeX~4.1 (and
\LaTeXe) in mansucripts prepared for submission to APS
journals. Further information can be found in the REV\TeX~4.1
documentation included in the distribution or available at
\url{http://authors.aps.org/revtex4/}.

When commands are referred to in this example file, they are always
shown with their required arguments, using normal \TeX{} format. In
this format, \verb+#1+, \verb+#2+, etc. stand for required
author-supplied arguments to commands. For example, in
\verb+\section{#1}+ the \verb+#1+ stands for the title text of the
author's section heading, and in \verb+\title{#1}+ the \verb+#1+
stands for the title text of the paper.

Line breaks in section headings at all levels can be introduced using
\textbackslash\textbackslash. A blank input line tells \TeX\ that the
paragraph has ended. Note that top-level section headings are
automatically uppercased. If a specific letter or word should appear in
lowercase instead, you must escape it using \verb+\lowercase{#1}+ as
in the word ``via'' above.

\subsection{\label{sec:level2}Second-level heading: Formatting}

This file may be formatted in either the \texttt{preprint} or
\texttt{reprint} style. \texttt{reprint} format mimics final journal output.
Either format may be used for submission purposes. \texttt{letter} sized paper should
be used when submitting to APS journals.

\subsubsection{Wide text (A level-3 head)}
The \texttt{widetext} environment will make the text the width of the
full page, as on page~\pageref{eq:wideeq}. (Note the use the
\verb+\pageref{#1}+ command to refer to the page number.)
\paragraph{Note (Fourth-level head is run in)}
The width-changing commands only take effect in two-column formatting.
There is no effect if text is in a single column.

\subsection{\label{sec:citeref}Citations and References}
A citation in text uses the command \verb+\cite{#1}+ or
\verb+\onlinecite{#1}+ and refers to an entry in the bibliography.
An entry in the bibliography is a reference to another document.

\subsubsection{Citations}
Because REV\TeX\ uses the \verb+natbib+ package of Patrick Daly,
the entire repertoire of commands in that package are available for your document;
see the \verb+natbib+ documentation for further details. Please note that
REV\TeX\ requires version 8.31a or later of \verb+natbib+.

\paragraph{Syntax}
The argument of \verb+\cite+ may be a single \emph{key},
or may consist of a comma-separated list of keys.
The citation \emph{key} may contain
letters, numbers, the dash (-) character, or the period (.) character.
New with natbib 8.3 is an extension to the syntax that allows for
a star (*) form and two optional arguments on the citation key itself.
The syntax of the \verb+\cite+ command is thus (informally stated)
\begin{quotation}\flushleft\leftskip1em
\verb+\cite+ \verb+{+ \emph{key} \verb+}+, or\\
\verb+\cite+ \verb+{+ \emph{optarg+key} \verb+}+, or\\
\verb+\cite+ \verb+{+ \emph{optarg+key} \verb+,+ \emph{optarg+key}\ldots \verb+}+,
\end{quotation}\noindent
where \emph{optarg+key} signifies
\begin{quotation}\flushleft\leftskip1em
\emph{key}, or\\
\texttt{*}\emph{key}, or\\
\texttt{[}\emph{pre}\texttt{]}\emph{key}, or\\
\texttt{[}\emph{pre}\texttt{]}\texttt{[}\emph{post}\texttt{]}\emph{key}, or even\\
\texttt{*}\texttt{[}\emph{pre}\texttt{]}\texttt{[}\emph{post}\texttt{]}\emph{key}.
\end{quotation}\noindent
where \emph{pre} and \emph{post} is whatever text you wish to place
at the beginning and end, respectively, of the bibliographic reference
(see Ref.~[\onlinecite{witten2001}] and the two under Ref.~[\onlinecite{feyn54}]).
(Keep in mind that no automatic space or punctuation is applied.)
It is highly recommended that you put the entire \emph{pre} or \emph{post} portion
within its own set of braces, for example:
\verb+\cite+ \verb+{+ \texttt{[} \verb+{+\emph{text}\verb+}+\texttt{]}\emph{key}\verb+}+.
The extra set of braces will keep \LaTeX\ out of trouble if your \emph{text} contains the comma (,) character.

The star (*) modifier to the \emph{key} signifies that the reference is to be
merged with the previous reference into a single bibliographic entry,
a common idiom in APS and AIP articles (see below, Ref.~[\onlinecite{epr}]).
When references are merged in this way, they are separated by a semicolon instead of
the period (full stop) that would otherwise appear.

\paragraph{Eliding repeated information}
When a reference is merged, some of its fields may be elided: for example,
when the author matches that of the previous reference, it is omitted.
If both author and journal match, both are omitted.
If the journal matches, but the author does not, the journal is replaced by \emph{ibid.},
as exemplified by Ref.~[\onlinecite{epr}].
These rules embody common editorial practice in APS and AIP journals and will only
be in effect if the markup features of the APS and AIP Bib\TeX\ styles is employed.

\paragraph{The options of the cite command itself}
Please note that optional arguments to the \emph{key} change the reference in the bibliography,
not the citation in the body of the document.
For the latter, use the optional arguments of the \verb+\cite+ command itself:
\verb+\cite+ \texttt{*}\allowbreak
\texttt{[}\emph{pre-cite}\texttt{]}\allowbreak
\texttt{[}\emph{post-cite}\texttt{]}\allowbreak
\verb+{+\emph{key-list}\verb+}+.

\subsubsection{Example citations}
By default, citations are numerical\cite{Beutler1994}.
Author-year citations are used when the journal is RMP.
To give a textual citation, use \verb+\onlinecite{#1}+:
Refs.~\onlinecite{[][{, and references therein}]witten2001,Bire82}.
By default, the \texttt{natbib} package automatically sorts your citations into numerical order and ``compresses'' runs of three or more consecutive numerical citations.
REV\TeX\ provides the ability to automatically change the punctuation when switching between journal styles that provide citations in square brackets and those that use a superscript style instead. This is done through the \texttt{citeautoscript} option. For instance, the journal style \texttt{prb} automatically invokes this option because \textit{Physical
Review B} uses superscript-style citations. The effect is to move the punctuation, which normally comes after a citation in square brackets, to its proper position before the superscript.
To illustrate, we cite several together
\cite{[See the explanation of time travel in ]feyn54,*[The classical relativistic treatment of ][ is a relative classic]epr,witten2001,Berman1983,Davies1998,Bire82},
and once again in different order (Refs.~\cite{epr,feyn54,Bire82,Berman1983,witten2001,Davies1998}).
Note that the citations were both compressed and sorted. Futhermore, running this sample file under the \texttt{prb} option will move the punctuation to the correct place.

When the \verb+prb+ class option is used, the \verb+\cite{#1}+ command
displays the reference's number as a superscript rather than in
square brackets. Note that the location of the \verb+\cite{#1}+
command should be adjusted for the reference style: the superscript
references in \verb+prb+ style must appear after punctuation;
otherwise the reference must appear before any punctuation. This
sample was written for the regular (non-\texttt{prb}) citation style.
The command \verb+\onlinecite{#1}+ in the \texttt{prb} style also
displays the reference on the baseline.

\subsubsection{References}
A reference in the bibliography is specified by a \verb+\bibitem{#1}+ command
with the same argument as the \verb+\cite{#1}+ command.
\verb+\bibitem{#1}+ commands may be crafted by hand or, preferably,
generated by Bib\TeX.
REV\TeX~4.1 includes Bib\TeX\ style files
\verb+apsrev4-1.bst+, \verb+apsrmp4-1.bst+ appropriate for
\textit{Physical Review} and \textit{Reviews of Modern Physics},
respectively. To display titles for cited journal articles, use the \texttt{longbibliography} class option.

\subsubsection{Example references}
This sample file employs the \verb+\bibliography+ command,
which formats the \texttt{\jobname .bbl} file
and specifies which bibliographic databases are to be used by Bib\TeX\
(one of these should be by arXiv convention \texttt{\jobname .bib}).
Running Bib\TeX\ (via \texttt{bibtex \jobname})
after the first pass of \LaTeX\ produces the file
\texttt{\jobname .bbl} which contains the automatically formatted
\verb+\bibitem+ commands (including extra markup information via
\verb+\bibinfo+ and \verb+\bibfield+ commands).
If not using Bib\TeX, you will have to create the \verb+thebibiliography+ environment
and its \verb+\bibitem+ commands by hand.

Numerous examples of the use of the APS bibliographic entry types appear in the bibliography of this sample document.
You can refer to the \texttt{\jobname .bib} file,
and compare its information to the formatted bibliography itself.

\subsection{Footnotes}%
Footnotes, produced using the \verb+\footnote{#1}+ command,
usually integrated into the bibliography alongside the other entries.
Numerical citation styles do this%
\footnote{Automatically placing footnotes into the bibliography requires using BibTeX to compile the bibliography.};
author-year citation styles place the footnote at the bottom of the text column.
Note: due to the method used to place footnotes in the bibliography,
\emph{you must re-run Bib\TeX\ every time you change any of your document's footnotes}.

\section{Math and Equations}
Inline math may be typeset using the \verb+$+ delimiters. Bold math
symbols may be achieved using the \verb+bm+ package and the
\verb+\bm{#1}+ command it supplies. For instance, a bold $\alpha$ can
be typeset as \verb+$\bm{\alpha}$+ giving $\bm{\alpha}$. Fraktur and
Blackboard (or open face or double struck) characters should be
typeset using the \verb+\mathfrak{#1}+ and \verb+\mathbb{#1}+ commands
respectively. Both are supplied by the \texttt{amssymb} package. For
example, \verb+$\mathbb{R}$+ gives $\mathbb{R}$ and
\verb+$\mathfrak{G}$+ gives $\mathfrak{G}$

In \LaTeX\ there are many different ways to display equations, and a
few preferred ways are noted below. Displayed math will center by
default. Use the class option \verb+fleqn+ to flush equations left.

Below we have numbered single-line equations; this is the most common
type of equation in \textit{Physical Review}:
\begin{eqnarray}
\chi_+(p)\alt{\bf [}2|{\bf p}|(|{\bf p}|+p_z){\bf ]}^{-1/2}
\left(
\begin{array}{c}
|{\bf p}|+p_z\\
px+ip_y
\end{array}\right)\;,
\\
\left\{%
 \openone234567890abc123\alpha\beta\gamma\delta1234556\alpha\beta
 \frac{1\sum^{a}_{b}}{A^2}%
\right\}%
\label{eq:one}.
\end{eqnarray}
Note the open one in Eq.~(\ref{eq:one}).

Not all numbered equations will fit within a narrow column this
way. The equation number will move down automatically if it cannot fit
on the same line with a one-line equation:
\begin{equation}
\left\{
 ab12345678abc123456abcdef\alpha\beta\gamma\delta1234556\alpha\beta
 \frac{1\sum^{a}_{b}}{A^2}%
\right\}.
\end{equation}

When the \verb+\label{#1}+ command is used [cf. input for
Eq.~(\ref{eq:one})], the equation can be referred to in text without
knowing the equation number that \TeX\ will assign to it. Just
use \verb+\ref{#1}+, where \verb+#1+ is the same name that used in
the \verb+\label{#1}+ command.

Unnumbered single-line equations can be typeset
using the \verb+\[+, \verb+\]+ format:
\[g^+g^+ \rightarrow g^+g^+g^+g^+ \dots ~,~~q^+q^+\rightarrow
q^+g^+g^+ \dots ~. \]

\subsection{Multiline equations}

Multiline equations are obtained by using the \verb+eqnarray+
environment.  Use the \verb+\nonumber+ command at the end of each line
to avoid assigning a number:
\begin{eqnarray}
{\cal M}=&&ig_Z^2(4E_1E_2)^{1/2}(l_i^2)^{-1}
\delta_{\sigma_1,-\sigma_2}
(g_{\sigma_2}^e)^2\chi_{-\sigma_2}(p_2)\nonumber\\
&&\times
[\epsilon_jl_i\epsilon_i]_{\sigma_1}\chi_{\sigma_1}(p_1),
\end{eqnarray}
\begin{eqnarray}
\sum \vert M^{\text{viol}}_g \vert ^2&=&g^{2n-4}_S(Q^2)~N^{n-2}
        (N^2-1)\nonumber \\
 & &\times \left( \sum_{i<j}\right)
  \sum_{\text{perm}}
 \frac{1}{S_{12}}
 \frac{1}{S_{12}}
 \sum_\tau c^f_\tau~.
\end{eqnarray}
\textbf{Note:} Do not use \verb+\label{#1}+ on a line of a multiline
equation if \verb+\nonumber+ is also used on that line. Incorrect
cross-referencing will result. Notice the use \verb+\text{#1}+ for
using a Roman font within a math environment.

To set a multiline equation without \emph{any} equation
numbers, use the \verb+\begin{eqnarray*}+,
\verb+\end{eqnarray*}+ format:
\begin{eqnarray*}
\sum \vert M^{\text{viol}}_g \vert ^2&=&g^{2n-4}_S(Q^2)~N^{n-2}
        (N^2-1)\\
 & &\times \left( \sum_{i<j}\right)
 \left(
  \sum_{\text{perm}}\frac{1}{S_{12}S_{23}S_{n1}}
 \right)
 \frac{1}{S_{12}}~.
\end{eqnarray*}

To obtain numbers not normally produced by the automatic numbering,
use the \verb+\tag{#1}+ command, where \verb+#1+ is the desired
equation number. For example, to get an equation number of
(\ref{eq:mynum}),
\begin{equation}
g^+g^+ \rightarrow g^+g^+g^+g^+ \dots ~,~~q^+q^+\rightarrow
q^+g^+g^+ \dots ~. \tag{2.6$'$}\label{eq:mynum}
\end{equation}

\paragraph{A few notes on \texttt{tag}s}
\verb+\tag{#1}+ requires the \texttt{amsmath} package.
Place the \verb+\tag{#1}+ command before the \verb+\label{#1}+, if any.
The numbering produced by \verb+\tag{#1}+ \textit{does not affect}
the automatic numbering in REV\TeX;
therefore, the number must be known ahead of time,
and it must be manually adjusted if other equations are added.
\verb+\tag{#1}+ works with both single-line and multiline equations.
\verb+\tag{#1}+ should only be used in exceptional cases---%
do not use it to number many equations in your paper.
Please note that this feature of the \texttt{amsmath} package
is \emph{not} compatible with the \texttt{hyperref} (6.77u) package.

Enclosing display math within
\verb+\begin{subequations}+ and \verb+\end{subequations}+ will produce
a set of equations that are labeled with letters, as shown in
Eqs.~(\ref{subeq:1}) and (\ref{subeq:2}) below.
You may include any number of single-line and multiline equations,
although it is probably not a good idea to follow one display math
directly after another.
\begin{subequations}
\label{eq:whole}
\begin{eqnarray}
{\cal M}=&&ig_Z^2(4E_1E_2)^{1/2}(l_i^2)^{-1}
(g_{\sigma_2}^e)^2\chi_{-\sigma_2}(p_2)\nonumber\\
&&\times
[\epsilon_i]_{\sigma_1}\chi_{\sigma_1}(p_1).\label{subeq:2}
\end{eqnarray}
\begin{equation}
\left\{
 abc123456abcdef\alpha\beta\gamma\delta1234556\alpha\beta
 \frac{1\sum^{a}_{b}}{A^2}
\right\},\label{subeq:1}
\end{equation}
\end{subequations}
Giving a \verb+\label{#1}+ command directly after the \verb+\begin{subequations}+,
allows you to reference all the equations in the \texttt{subequations} environment.
For example, the equations in the preceding subequations environment were
Eqs.~(\ref{eq:whole}).

\subsubsection{Wide equations}
The equation that follows is set in a wide format, i.e., it spans the full page.
The wide format is reserved for long equations
that cannot easily be set in a single column:
\begin{widetext}
\begin{equation}
{\cal R}^{(\text{d})}=
 g_{\sigma_2}^e
 \left(
   \frac{[\Gamma^Z(3,21)]_{\sigma_1}}{Q_{12}^2-M_W^2}
  +\frac{[\Gamma^Z(13,2)]_{\sigma_1}}{Q_{13}^2-M_W^2}
 \right)
 + x_WQ_e
 \left(
   \frac{[\Gamma^\gamma(3,21)]_{\sigma_1}}{Q_{12}^2-M_W^2}
  +\frac{[\Gamma^\gamma(13,2)]_{\sigma_1}}{Q_{13}^2-M_W^2}
 \right)\;.
 \label{eq:wideeq}
\end{equation}
\end{widetext}
This is typed to show how the output appears in wide format.
(Incidentally, since there is no blank line between the \texttt{equation} environment above
and the start of this paragraph, this paragraph is not indented.)

\section{Cross-referencing}
REV\TeX{} will automatically number such things as
sections, footnotes, equations, figure captions, and table captions.
In order to reference them in text, use the
\verb+\label{#1}+ and \verb+\ref{#1}+ commands.
To reference a particular page, use the \verb+\pageref{#1}+ command.

The \verb+\label{#1}+ should appear
within the section heading,
within the footnote text,
within the equation, or
within the table or figure caption.
The \verb+\ref{#1}+ command
is used in text at the point where the reference is to be displayed.
Some examples: Section~\ref{sec:level1} on page~\pageref{sec:level1},
Table~\ref{tab:table1},%
\begin{table}[b
\caption{\label{tab:table1}%
A table that fits into a single column of a two-column layout.
Note that REV\TeX~4 adjusts the intercolumn spacing so that the table fills the
entire width of the column. Table captions are numbered
automatically.
This table illustrates left-, center-, decimal- and right-aligned columns,
along with the use of the \texttt{ruledtabular} environment which sets the
Scotch (double) rules above and below the alignment, per APS style.
}
\begin{ruledtabular}
\begin{tabular}{lcdr}
\textrm{Left\footnote{Note a.}}&
\textrm{Centered\footnote{Note b.}}&
\multicolumn{1}{c}{\textrm{Decimal}}&
\textrm{Right}\\
\colrule
1 & 2 & 3.001 & 4\\
10 & 20 & 30 & 40\\
100 & 200 & 300.0 & 400\\
\end{tabular}
\end{ruledtabular}
\end{table}
and Fig.~\ref{fig:epsart}.%
\begin{figure}[b]
\includegraphics{fig_1
\caption{\label{fig:epsart} A figure caption. The figure captions are
automatically numbered.}
\end{figure}

\section{Floats: Figures, Tables, Videos, etc.}
Figures and tables are usually allowed to ``float'', which means that their
placement is determined by \LaTeX, while the document is being typeset.

Use the \texttt{figure} environment for a figure, the \texttt{table} environment for a table.
In each case, use the \verb+\caption+ command within to give the text of the
figure or table caption along with the \verb+\label+ command to provide
a key for referring to this figure or table.
The typical content of a figure is an image of some kind;
that of a table is an alignment.%
\begin{figure*}
\includegraphics{fig_2
\caption{\label{fig:wide}Use the figure* environment to get a wide
figure that spans the page in \texttt{twocolumn} formatting.}
\end{figure*}
\begin{table*}
\caption{\label{tab:table3}This is a wide table that spans the full page
width in a two-column layout. It is formatted using the
\texttt{table*} environment. It also demonstates the use of
\textbackslash\texttt{multicolumn} in rows with entries that span
more than one column.}
\begin{ruledtabular}
\begin{tabular}{ccccc}
 &\multicolumn{2}{c}{$D_{4h}^1$}&\multicolumn{2}{c}{$D_{4h}^5$}\\
 Ion&1st alternative&2nd alternative&lst alternative
&2nd alternative\\ \hline
 K&$(2e)+(2f)$&$(4i)$ &$(2c)+(2d)$&$(4f)$ \\
 Mn&$(2g)$\footnote{The $z$ parameter of these positions is $z\sim\frac{1}{4}$.}
 &$(a)+(b)+(c)+(d)$&$(4e)$&$(2a)+(2b)$\\
 Cl&$(a)+(b)+(c)+(d)$&$(2g)$\footnotemark[1]
 &$(4e)^{\text{a}}$\\
 He&$(8r)^{\text{a}}$&$(4j)^{\text{a}}$&$(4g)^{\text{a}}$\\
 Ag& &$(4k)^{\text{a}}$& &$(4h)^{\text{a}}$\\
\end{tabular}
\end{ruledtabular}
\end{table*}

Insert an image using either the \texttt{graphics} or
\texttt{graphix} packages, which define the \verb+\includegraphics{#1}+ command.
(The two packages differ in respect of the optional arguments
used to specify the orientation, scaling, and translation of the image.)
To create an alignment, use the \texttt{tabular} environment.

The best place to locate the \texttt{figure} or \texttt{table} environment
is immediately following its first reference in text; this sample document
illustrates this practice for Fig.~\ref{fig:epsart}, which
shows a figure that is small enough to fit in a single column.

In exceptional cases, you will need to move the float earlier in the document, as was done
with Table~\ref{tab:table3}: \LaTeX's float placement algorithms need to know
about a full-page-width float earlier.

Fig.~\ref{fig:wide}
has content that is too wide for a single column,
so the \texttt{figure*} environment has been used.%
\begin{table}[b]
\caption{\label{tab:table4}%
Numbers in columns Three--Five are aligned with the ``d'' column specifier
(requires the \texttt{dcolumn} package).
Non-numeric entries (those entries without a ``.'') in a ``d'' column are aligned on the decimal point.
Use the ``D'' specifier for more complex layouts. }
\begin{ruledtabular}
\begin{tabular}{ccddd}
One&Two&
\multicolumn{1}{c}{\textrm{Three}}&
\multicolumn{1}{c}{\textrm{Four}}&
\multicolumn{1}{c}{\textrm{Five}}\\
\hline
one&two&\mbox{three}&\mbox{four}&\mbox{five}\\
He&2& 2.77234 & 45672. & 0.69 \\
C\footnote{Some tables require footnotes.}
  &C\footnote{Some tables need more than one footnote.}
  & 12537.64 & 37.66345 & 86.37 \\
\end{tabular}
\end{ruledtabular}
\end{table}

The content of a table is typically a \texttt{tabular} environment,
giving rows of type in aligned columns.
Column entries separated by \verb+&+'s, and
each row ends with \textbackslash\textbackslash.
The required argument for the \texttt{tabular} environment
specifies how data are aligned in the columns.
For instance, entries may be centered, left-justified, right-justified, aligned on a decimal
point.
Extra column-spacing may be be specified as well,
although REV\TeX~4 sets this spacing so that the columns fill the width of the
table. Horizontal rules are typeset using the \verb+\hline+
command. The doubled (or Scotch) rules that appear at the top and
bottom of a table can be achieved enclosing the \texttt{tabular}
environment within a \texttt{ruledtabular} environment. Rows whose
columns span multiple columns can be typeset using the
\verb+\multicolumn{#1}{#2}{#3}+ command (for example, see the first
row of Table~\ref{tab:table3}).%

Tables~\ref{tab:table1}, \ref{tab:table3}, \ref{tab:table4}, and \ref{tab:table2}%
\begin{table}[b]
\caption{\label{tab:table2}
A table with numerous columns that still fits into a single column.
Here, several entries share the same footnote.
Inspect the \LaTeX\ input for this table to see exactly how it is done.}
\begin{ruledtabular}
\begin{tabular}{cccccccc}
 &$r_c$ (\AA)&$r_0$ (\AA)&$\kappa r_0$&
 &$r_c$ (\AA) &$r_0$ (\AA)&$\kappa r_0$\\
\hline
Cu& 0.800 & 14.10 & 2.550 &Sn\footnotemark[1]
& 0.680 & 1.870 & 3.700 \\
Ag& 0.990 & 15.90 & 2.710 &Pb\footnotemark[2]
& 0.450 & 1.930 & 3.760 \\
Au& 1.150 & 15.90 & 2.710 &Ca\footnotemark[3]
& 0.750 & 2.170 & 3.560 \\
Mg& 0.490 & 17.60 & 3.200 &Sr\footnotemark[4]
& 0.900 & 2.370 & 3.720 \\
Zn& 0.300 & 15.20 & 2.970 &Li\footnotemark[2]
& 0.380 & 1.730 & 2.830 \\
Cd& 0.530 & 17.10 & 3.160 &Na\footnotemark[5]
& 0.760 & 2.110 & 3.120 \\
Hg& 0.550 & 17.80 & 3.220 &K\footnotemark[5]
&  1.120 & 2.620 & 3.480 \\
Al& 0.230 & 15.80 & 3.240 &Rb\footnotemark[3]
& 1.330 & 2.800 & 3.590 \\
Ga& 0.310 & 16.70 & 3.330 &Cs\footnotemark[4]
& 1.420 & 3.030 & 3.740 \\
In& 0.460 & 18.40 & 3.500 &Ba\footnotemark[5]
& 0.960 & 2.460 & 3.780 \\
Tl& 0.480 & 18.90 & 3.550 & & & & \\
\end{tabular}
\end{ruledtabular}
\footnotetext[1]{Here's the first, from Ref.~\onlinecite{feyn54}.}
\footnotetext[2]{Here's the second.}
\footnotetext[3]{Here's the third.}
\footnotetext[4]{Here's the fourth.}
\footnotetext[5]{And etc.}
\end{table}
show various effects.
A table that fits in a single column employs the \texttt{table}
environment.
Table~\ref{tab:table3} is a wide table, set with the \texttt{table*} environment.
Long tables may need to break across pages.
The most straightforward way to accomplish this is to specify
the \verb+[H]+ float placement on the \texttt{table} or
\texttt{table*} environment.
However, the \LaTeXe\ package \texttt{longtable} allows headers and footers to be specified for each page of the table.
A simple example of the use of \texttt{longtable} can be found
in the file \texttt{summary.tex} that is included with the REV\TeX~4
distribution.

There are two methods for setting footnotes within a table (these
footnotes will be displayed directly below the table rather than at
the bottom of the page or in the bibliography). The easiest
and preferred method is just to use the \verb+\footnote{#1}+
command. This will automatically enumerate the footnotes with
lowercase roman letters. However, it is sometimes necessary to have
multiple entries in the table share the same footnote. In this case,
there is no choice but to manually create the footnotes using
\verb+\footnotemark[#1]+ and \verb+\footnotetext[#1]{#2}+.
\texttt{\#1} is a numeric value. Each time the same value for
\texttt{\#1} is used, the same mark is produced in the table. The
\verb+\footnotetext[#1]{#2}+ commands are placed after the \texttt{tabular}
environment. Examine the \LaTeX\ source and output for
Tables~\ref{tab:table1} and \ref{tab:table2}
for examples.

Video~\ref{vid:PRSTPER.4.010101}
illustrates several features new with REV\TeX4.1,
starting with the \texttt{video} environment, which is in the same category with
\texttt{figure} and \texttt{table}.%
\begin{video}
\href{http://prst-per.aps.org/multimedia/PRSTPER/v4/i1/e010101/e010101_vid1a.mpg}{\includegraphics{vid_1a}}%
 \quad
\href{http://prst-per.aps.org/multimedia/PRSTPER/v4/i1/e010101/e010101_vid1b.mpg}{\includegraphics{vid_1b}}
 \setfloatlink{http://link.aps.org/multimedia/PRSTPER/v4/i1/e010101}%
 \caption{\label{vid:PRSTPER.4.010101}%
  Students explain their initial idea about Newton's third law to a teaching assistant.
  Clip (a): same force.
  Clip (b): move backwards.
 }%
\end{video}
The \verb+\setfloatlink+ command causes the title of the video to be a hyperlink to the
indicated URL; it may be used with any environment that takes the \verb+\caption+
command.
The \verb+\href+ command has the same significance as it does in the context of
the \texttt{hyperref} package: the second argument is a piece of text to be
typeset in your document; the first is its hyperlink, a URL.

\textit{Physical Review} style requires that the initial citation of
figures or tables be in numerical order in text, so don't cite
Fig.~\ref{fig:wide} until Fig.~\ref{fig:epsart} has been cited.

\begin{acknowledgments}
We wish to acknowledge the support of the author community in using
REV\TeX{}, offering suggestions and encouragement, testing new versions,
\dots.
\end{acknowledgments}

\section{}
\subsection{}
\subsubsection{}