text:
\section{Introduction}

Originally posited in the context of $^3$He by Berezinskii\cite{Berezinskii1974} and then later for superconductivity\cite{BalatskyPRB1992} odd-frequency (odd-$\omega$) pairing offers a theoretical means for interacting Fermions to avoid strong on-site repulsion by pairing Fermions at different times. An intriguing consequence of odd-$\omega$ pairing is that it introduces alternatives to the conventional classification of Fermionic pairs as either spin triplet odd-parity or spin singlet even-parity. Given the inherently dynamical nature of odd-$\omega$ pairing it seems natural to discuss the possibility of its emergence in systems that possess explicit time-dependence, however we are unaware of any previous study with this focus. The purpose of this paper is to elucidate the fact that odd-$\omega$ superconductivity is a concept generalizable to driven systems, even in the absence of translation invariance and to discuss the conditions under which coupling a conventional superconductor to a time-dependent potential should give rise to odd-$\omega$ pair amplitudes. Additionally, we demonstrate explicitly that in the presence of a time-dependent and spatially nonuniform electric field a conventional superconductor will develop odd-$\omega$ pair amplitudes. The emergence of these odd-$\omega$ amplitudes is then shown to be related to phenomena measurable using scanning tunneling microscopy (STM) and photoemission spectroscopy.

Previous research on odd-$\omega$ pairing has been devoted to the study of the thermodynamic stability of odd-$\omega$ phases\cite{heid1995thermodynamic,solenov2009thermodynamical,kusunose2011puzzle} as well as the identification of systems in which odd-$\omega$ pairing could be induced including: ferromagnetic - superconductor heterostructures \cite{BergeretPRL2001,bergeret2005odd,yokoyama2007manifestation,houzet2008ferromagnetic,EschrigNat2008,LinderPRB2008,crepin2015odd} with recent experimental indications of its realization at the interface of Nb thin film and epitaxial Ho\cite{di2015signature}, certain kinds of topological insulator - superconductor systems \cite{YokoyamaPRB2012,Black-SchafferPRB2012,Black-SchafferPRB2013,TriolaPRB2014}, certain normal metal - superconductor junctions due to broken symmetries\cite{tanaka2007theory,TanakaPRB2007,LinderPRL2009,LinderPRB2010_2,TanakaJPSJ2012}, and multiband superconductors due to finite interband hybridization\cite{black2013odd,komendova2015experimentally}. Additional work has investigated the emergence of proximity-induced odd-$\omega$ pairing in generic two-dimensional bilayers coupled to conventional $s$-wave superconductors\cite{parhizgar_2014_prb} and in two-dimensional electron gases coupled to superconductor thin films\cite{triola2016}. Furthermore, the concept of odd-$\omega$ order parameters can be generalized to charge and spin density waves\cite{pivovarov2001odd,kedem2015odd} and Majorana fermions\cite{huang2015odd}, demonstrating the ubiquity of the odd-$\omega$ class of ordered states.

\section{Odd-frequency Pairing in Driven Systems}

To establish the relevance of odd-$\omega$ pairing in driven systems we start by noting that the anomalous Green's function of a superconductor can be defined as
$F_{\sigma_1,\alpha_1;\sigma_2,\alpha_2}(\textbf{r}_1,t_1;\textbf{r}_2,t_2)=-i\langle T \psi_{\sigma_1,\alpha_1,\textbf{r}_1,t_1}\psi_{\sigma_2,\alpha_2,\textbf{r}_2,t_2}\rangle$
where $T$ is the time-ordering operator, and $\psi_{\sigma,\alpha,\textbf{r},t}$ annihilates an electron at time $t$, position $\textbf{r}$, with spin $\sigma$, and orbital degrees of freedom described by $\alpha$. Using the definition of time-ordering for Fermions it is straightforward to show that: $F_{\sigma_1,\alpha_1;\sigma_2,\alpha_2}(\textbf{r}_1,t_1;\textbf{r}_2,t_2)=-F_{\sigma_2,\alpha_2;\sigma_1,\alpha_1}(\textbf{r}_2,t_2;\textbf{r}_1,t_1)$. This relation tells us that the wavefunction describing the Cooper pairs, $\Psi$, must obey $\mathcal{S}\mathcal{T}\mathcal{O}\mathcal{P}\Psi=-\Psi$ where: $\mathcal{S}$ acts on spin ($\sigma_1\leftrightarrow\sigma_2$); $\mathcal{T}$ reverses time coordinates ($t_1\leftrightarrow t_2$); $\mathcal{O}$ interchanges orbital degrees of freedom ($\alpha_1\leftrightarrow\alpha_2$); and $\mathcal{P}$ is the spatial parity operator ($\textbf{r}_1\leftrightarrow\textbf{r}_2$). Using this property of $\Psi$ together with the fact that all four transformations are involutory, the possible symmetries of the Cooper pair wavefunction may be divided into 8 different classes based on how they transform under $\mathcal{S}$, $\mathcal{T}$, $\mathcal{O}$, and $\mathcal{P}$, see Table~\ref{table:classification}.
\begin{center}
\begin{table}[htb]
\begin{tabular}{c || c | c | c | c | c | c | c | c}
Spin & -1 & 1  & 1  & 1  & 1  & -1 & -1 & -1 \\
\hline
Time & 1  & 1  & 1  & -1 & -1 & -1 & -1 & 1 \\
\hline
Orbital & 1  & 1  & -1 & 1  & -1 & -1 & 1  & -1 \\
\hline
Parity & 1  & -1 & 1  & 1  & -1 & 1  & -1 & -1
\end{tabular}
\caption{Classification of the 8 allowed symmetry classes for the superconducting order parameter (Cooper pair wavefunction).}
\label{table:classification}
\end{table}
\end{center}

Fourier transforming the anomalous Green's function with respect to the average and relative position and time it can be shown that:
\begin{equation}
F_{\sigma_1,\alpha_1;\sigma_2,\alpha_2}(\textbf{k},\omega;\textbf{Q},\Omega) =-F_{\sigma_2,\alpha_2;\sigma_1,\alpha_1}(-\textbf{k},-\omega;\textbf{Q},\Omega)
\label{eq:bere}
\end{equation}
where $\textbf{k}$ and $\omega$ ($\textbf{Q}$ and $\Omega$) are the momentum and frequency conjugate to the relative (average) position and time coordinates. Eq.~(\ref{eq:bere}) encapsulates the generalization of the Berezinskii classification to systems which lack translation symmetry in space and time. We note that there is no requirement on the symmetry with respect to the average momenta or frequency. Thus if the dependence on $\textbf{Q}$ and $\Omega$ is suppressed this reduces to the standard Berezinskii classification.

With a framework now established to classify the symmetries of a driven superconductor we would like to study a concrete example of a driven system with pair amplitudes falling into one of the odd-$\omega$ classes. Heuristically, one might expect that subjecting a conventional superconductor to a time-dependent potential could lead to the creation of odd-$\omega$ pair amplitudes since the correlation functions will then develop a nontrivial dependence on the absolute time arguments through convolutions with the driving potential. This time-dependence will then manifest itself as additional pair amplitudes that must each fall into one of the 8 symmetry classes outlined in Table~\ref{table:classification} and, a priori, there is no reason to exclude the possible emergence of odd-$\omega$ terms. However, we note that if the driving potential acts trivially on orbital and spin degrees of freedom then, in the case where the original superconductor is a spin singlet, and even in both parity and time ($\mathcal{S}=-1$, $\mathcal{T}=1$, $\mathcal{O}=1$, $\mathcal{P}=1$), the only possible terms that are odd-$\omega$ would be in the class defined by ($\mathcal{S}=-1$, $\mathcal{T}=-1$, $\mathcal{O}=1$, $\mathcal{P}=-1$). Therefore, we expect that the driving potential should also depend on position to generate odd-parity amplitudes. Alternatively, if the drive were allowed to act nontrivially on spin or orbital degrees of freedom it should be possible to generate odd-$\omega$ terms in the class defined by ($\mathcal{S}=1$, $\mathcal{T}=-1$, $\mathcal{O}=1$, $\mathcal{P}=1$) or ($\mathcal{S}=-1$, $\mathcal{T}=-1$, $\mathcal{O}=-1$, $\mathcal{P}=1$), respectively. We will now demonstrate this transmutation of symmetries explicitly for the case of a conventional superconductor driven by a spatially nonuniform AC electric field.
\begin{figure}
 \begin{center}
  \centering
  \includegraphics[width=0.5\textwidth]{Schematic_Fig_4.eps}
  \caption{(color online) Schematic of a driven superconducting system with a 2D superconducting region described by Eq.~(\ref{eq:Ham}) lying between two insulating slabs each capped by a conducting electrode configured in such a way as to generate a spatially nonuniform electric field. The AC voltage acts as a time-dependent drive. Such a device could be realized by sandwiching a thin film superconductor, like Pb\cite{zhang2005band,eom2006persistent,brun2009reduction,zhang2010superconductivity}, between two insulating wafers.
         }
  \label{fig:schematic}
 \end{center}
\end{figure}

\subsection{Model for Driven Superconductor}

The schematic shown in Fig.~\ref{fig:schematic} represents a conceptually simple method for coupling a two-dimensional (2D) $s$-wave singlet superconductor to a time- and position-dependent drive. This system could be realized using a thin film superconductor, for example Pb\cite{zhang2005band,eom2006persistent,brun2009reduction,zhang2010superconductivity}, sandwiched between two insulators and the drive could be implemented by biasing the sample with a spatially nonuniform AC voltage. As we will show, in such a device we expect the Cooper pairs in the 2D superconductor to develop odd-$\omega$ correlations. The superconducting region in Fig.~\ref{fig:schematic} is described by the Hamiltonian $H=H_0+H_{i}+H_t$ where:
\begin{equation}
\begin{aligned}
H_0 &= \sum_{\textbf{r},\textbf{r}',\sigma}(\gamma_{\textbf{r}\textbf{r}'}-\mu \delta_{\textbf{r}\textbf{r}'} ) \psi^\dagger_{\sigma,\textbf{r}} \psi_{\sigma,\textbf{r}'} \\
H_i&= \sum_{\textbf{r}}\lambda \psi^\dagger_{\uparrow,\textbf{r}}\psi^\dagger_{\downarrow,\textbf{r}}\psi_{\downarrow,\textbf{r}} \psi_{\uparrow,\textbf{r}} \\
H_t &=\sum_{\textbf{r},\sigma} U_{\textbf{r},t}\psi^\dagger_{\sigma,\textbf{r}} \psi_{\sigma,\textbf{r}}
\end{aligned}
\label{eq:Ham}
\end{equation}
where $\psi^\dagger_{\sigma,\textbf{r}}$ ($\psi_{\sigma,\textbf{r}}$) creates (annihilates) a fermionic quasiparticle state with spin $\sigma$ on site $\textbf{r}$, $\gamma_{\textbf{r}\textbf{r}'}$ is the hopping integral describing the tight-binding model of the normal state quasiparticles with chemical potential $\mu$, $\lambda$ describes an attractive interaction between quasiparticles, and $U_{\textbf{r},t}$ is a time- and position-dependent potential modeling the effect of the AC voltage.

To study the case of a time-dependent drive, $U_{\textbf{r},t}$, we employ non-equilibrium Green's functions\cite{rammer2007quantum,stefanucci2013nonequilibrium}. In this formalism the Green's function can be represented as an 8$\times$8 matrix written in a combined Keldysh$\times$particle-hole$\times$spin space:
\begin{equation}
\hat{\mathcal{G}}(x_1;x_2)=\left(
\begin{array}{cc}
\hat{\mathcal{G}}^{R}(x_1;x_2) & \hat{\mathcal{G}}^{K}(x_1;x_2) \\
0 & \hat{\mathcal{G}}^{A}(x_1;x_2)
\end{array} \right)
\label{eq:g_keldysh}
\end{equation}
where we adopt the notation $x_i\equiv(\textbf{r}_i,t_i)$ where $\textbf{r}_i$ is a position in the 2D superconductor and $t_i$ is a time defined on the interval $(-\infty,\infty)$, and where each component, $\hat{\mathcal{G}}^{\alpha}(x_1;x_2)$, is a matrix in particle-hole$\times$spin space:
\begin{equation}
\hat{\mathcal{G}}^{\alpha}(x_1;x_2)=\left(
\begin{array}{cc}
\hat{G}^{\alpha}(x_1;x_2) & \hat{F}^{\alpha}(x_1;x_2) \\
\hat{\overline{F}}^{\alpha}(x_1;x_2) & \hat{\overline{G}}^{\alpha}(x_1;x_2)
\end{array} \right)
\label{eq:g_nambu}
\end{equation}
where the ``hat" indicates that each block is a 2$\times$2 matrix in spin space. The retarded and Keldysh entries are defined as:
\begin{equation}
\begin{aligned}
G^{R}_{\sigma_1 \sigma_2}(x_1;x_2) &= -i\theta(t_1-t_2)\langle \{\psi_{\sigma_1,\textbf{r}_1}(t_1),\psi^\dagger_{\sigma_2,\textbf{r}_2}(t_2) \}\rangle \\
F^{R}_{\sigma_1 \sigma_2}(x_1;x_2) &= -i\theta(t_1-t_2)\langle \{\psi_{\sigma_1,\textbf{r}_1}(t_1),\psi_{\sigma_2,\textbf{r}_2}(t_2) \}\rangle \\
\overline{G}^{R}_{\sigma_1 \sigma_2}(x_1;x_2) &= -i\theta(t_1-t_2)\langle \{\psi^\dagger_{\sigma_1,\textbf{r}_1}(t_1),\psi_{\sigma_2,\textbf{r}_2}(t_2) \}\rangle \\
\overline{F}^{R}_{\sigma_1 \sigma_2}(x_1;x_2) &= -i\theta(t_1-t_2)\langle \{\psi^\dagger_{\sigma_1,\textbf{r}_1}(t_1),\psi^\dagger_{\sigma_2,\textbf{r}_2}(t_2) \}\rangle \\
G^{K}_{\sigma_1 \sigma_2}(x_1;x_2) &= -i\langle [\psi_{\sigma_1,\textbf{r}_1}(t_1),\psi^\dagger_{\sigma_2,\textbf{r}_2}(t_2) ]\rangle \\
F^{K}_{\sigma_1 \sigma_2}(x_1;x_2) &= -i\langle [\psi_{\sigma_1,\textbf{r}_1}(t_1),\psi_{\sigma_2,\textbf{r}_2}(t_2) ]\rangle \\
\overline{G}^{K}_{\sigma_1 \sigma_2}(x_1;x_2) &= -i\langle [\psi^\dagger_{\sigma_1,\textbf{r}_1}(t_1),\psi_{\sigma_2,\textbf{r}_2}(t_2) ]\rangle \\
\overline{F}^{K}_{\sigma_1 \sigma_2}(x_1;x_2) &= -i\langle [\psi^\dagger_{\sigma_1,\textbf{r}_1}(t_1),\psi^\dagger_{\sigma_2,\textbf{r}_2}(t_2) ]\rangle
\end{aligned}
\label{eq:g_retkel}
\end{equation}
while the advanced block is related to the retarded block by $\hat{\mathcal{G}}^{A}(x_1;x_2)=\hat{\mathcal{G}}^{R}(x_2;x_1)^\dagger$.

To describe the superconducting state we decouple the four-fermion term in Eq.(\ref{eq:Ham}) using mean field theory. In this case we find that $\lambda\psi^\dagger_{\uparrow,\textbf{r}}\psi^\dagger_{\downarrow,\textbf{r}}\psi_{\downarrow,\textbf{r}} \psi_{\uparrow,\textbf{r}}\approx\Delta(\textbf{r},t)\psi^\dagger_{\uparrow,\textbf{r}}\psi^\dagger_{\downarrow,\textbf{r}}+\text{h.c.}$ where $\Delta(\textbf{r},t)=\lambda\langle \psi_{\downarrow,\textbf{r}}(t)\psi_{\uparrow,\textbf{r}}(t)\rangle$. Written in terms of the Green's functions defined in Eqs (\ref{eq:g_keldysh}) and (\ref{eq:g_nambu}), and making the position- and time-dependence explicit, we have:
\begin{equation}
\Delta(\textbf{r},t)=i\frac{\lambda}{2}\left( F^{R}_{\downarrow\uparrow}(\textbf{r},t;\textbf{r},t) - F^{A}_{\downarrow\uparrow}(\textbf{r},t;\textbf{r},t) + F^{K}_{\downarrow\uparrow}(\textbf{r},t;\textbf{r},t) \right).
\label{eq:gap}
\end{equation}

With the mean field defined in Eq (\ref{eq:gap}) and using the Heisenberg equations of motion for the operators $\psi_{\sigma,\textbf{r}}(t)$, one can show that the equations of motion for the Green's functions in Eq (\ref{eq:g_keldysh}) are:
\begin{widetext}
\begin{equation}
\begin{aligned}
\hat{\mathcal{G}}^{-1}(x_1)\hat{\mathcal{G}}(x_1;x_2)&=\delta(x_1-x_2) \ \hat{\sigma}_0\otimes\hat{\rho}_0 \otimes\hat{\tau}_0 ; \\
\hat{\mathcal{G}}^{-1}(x_1) &= \left( \begin{array}{cc}
i\dfrac{d}{dt_1} - \left(-\dfrac{\nabla^2_{\textbf{r}_1}}{2m} -\mu-U_{\textbf{r}_1,t_1} \right) & -\Delta(\textbf{r}_1,t_1)i\hat{\sigma}_2 \\
-(\Delta(\textbf{r}_1,t_1)i\hat{\sigma}_2)^\dagger & i\dfrac{d}{dt_1} + \left(-\dfrac{\nabla^2_{\textbf{r}_1}}{2m} -\mu-U_{\textbf{r}_1,t_1} \right)\end{array}\right)\otimes \hat{\tau}_0
\label{eom}
\end{aligned}
\end{equation}
\end{widetext}
where $\hat{\sigma}_0$, $\hat{\rho}_0$, and $\hat{\tau}_0$ are the identity matrices in spin, particle-hole, and Keldysh space respectively, $\hat{\sigma}_{i\neq0}$ are the Pauli matrices in spin space, and we have approximated the hopping integral $\gamma_{\textbf{r},\textbf{r}'}$ by $\nabla^2_{\textbf{r}_1}/2m$.

In the absence of a drive ($U_{\textbf{r},t}=0$) the system is in equilibrium and translation invariant, therefore $\Delta(\textbf{r},t)=\Delta_0$ is constant. In this unperturbed case one can Fourier transform Eq (\ref{eom}) and find:
\begin{widetext}
\begin{equation}
\begin{aligned}
\hat{\mathcal{G}}_0^R(\textbf{k},\omega)&= \lim_{\eta \rightarrow 0^+}\frac{1}{(\omega+i\eta)^2-\xi^2_{\textbf{k}}-|\Delta_0|^2}\left( \begin{array}{cc}
\left( \omega +i\eta + \xi_\textbf{k} \right)\hat{\sigma}_0 & \Delta_0 i\hat{\sigma}_2 \\
-\Delta_0^{*} i\hat{\sigma}_2 & \left( \omega +i\eta - \xi_\textbf{k} \right)\hat{\sigma}_0
\end{array} \right) \\
\hat{\mathcal{G}}_0^A(\textbf{k},\omega)&= \lim_{\eta \rightarrow 0^+}\frac{1}{(\omega-i\eta)^2-\xi^2_{\textbf{k}}-|\Delta_0|^2}\left( \begin{array}{cc}
\left( \omega -i\eta + \xi_\textbf{k} \right)\hat{\sigma}_0 & \Delta_0 i\hat{\sigma}_2 \\
-\Delta_0^{*} i\hat{\sigma}_2 & \left( \omega -i\eta - \xi_\textbf{k} \right)\hat{\sigma}_0
\end{array} \right) \\
\hat{\mathcal{G}}_0^K(\textbf{k},\omega)&=\frac{-i\pi\tanh\left(\frac{\omega\beta}{2}\right)}{\sqrt{\xi_{\textbf{k}}^2 + |\Delta_0|^2}} \left[ \delta\left(\omega-\sqrt{\xi_\textbf{k}^2 +|\Delta_0|^2}\right) - \delta\left(\omega+\sqrt{\xi_\textbf{k}^2 +|\Delta_0|^2}\right)\right] \left( \begin{array}{cc}
\left(\omega+\xi_\textbf{k}\right)\hat{\sigma}_0 & \Delta_0 i\hat{\sigma}_2 \\
-\Delta_0^{*} i\hat{\sigma}_2 & \left(\omega -\xi_\textbf{k} \right)\hat{\sigma}_0
\end{array} \right)
\end{aligned}
\label{eq:rak}
\end{equation}
\end{widetext}
where $\xi_\textbf{k}=\frac{\hbar^2}{2m}k^2-\mu$ and $\beta$ is the inverse temperature of the superconductor.

In the presence of a time-dependent drive, $U_{\textbf{r},t}$, the Green's functions in Eq (\ref{eom}) can be related to the solutions to the undriven problem by the Dyson equation:
\begin{equation}
\hat{\mathcal{G}}(x_1;x_2)=\hat{\mathcal{G}}_0(x_1;x_2) - \int dx \hat{\mathcal{G}}_0(x_1;x) \hat{U}_{x} \hat{\mathcal{G}}(x;x_2) \\
\label{eq:dyson}
\end{equation}
where $\int dx=\int dt \int d\textbf{r}$, $\hat{U}_{x}=U_{\textbf{r},t} \ \hat{\sigma}_0\otimes\hat{\rho}_3\otimes\hat{\tau}_0$, and $\hat{\rho}_{3}$ is the Pauli matrix in particle-hole space.

For a strong drive, $U_{\textbf{r},t}$, Eq (\ref{eq:dyson}) must be solved with a self-consistent solution for $\Delta(\textbf{r},t)$ in which case Eq (\ref{eq:rak}) would no longer be valid. However, as shown in the appendix, the first order self-consistent corrections to $\Delta(\textbf{r},t)$ are given by:
\begin{equation}
\frac{\delta\Delta(\textbf{r},t)}{\Delta_0}=\frac{-\lambda N_0}{4}\frac{U_0}{\mu} e^{-i\textbf{Q}_0\cdot\textbf{r}-i\Omega_0 t} \left[2+\frac{\Omega_0}{2\mu}+\frac{\pi^2}{12}\overline{Q}_0^2\frac{\Delta_0^2}{\mu^2} \right]
\label{eq:delta_delta}
\end{equation}
where $N_0=m/2\pi\hbar^2$ is the density of states, $\overline{Q}_0$ is the driving wavevector times the coherence length $\xi=\hbar v_F/\pi\Delta_0$, $\Omega_0$ is the driving frequency, and $U_0$ is the amplitude of the drive.

We note that in superconducting Pb thin films typical values for these parameters are: $\lambda N_0\approx 1$\cite{zhang2005band,zhang2010superconductivity}, $\mu\approx 1.2$eV\cite{zhang2005band}, $\Delta_0\approx 1.17$meV\cite{eom2006persistent,brun2009reduction}, and $\xi\approx 49$nm\cite{zhang2010superconductivity}. Thus, for $\Omega_0<\mu$ ($<290 \text{THz}$) and $Q_0<\xi^{-1}$ ($<2\times 10^{-2}\text{nm}^{-1}$), $U_0$ can be as large as 100 meV and still provide adequate suppression of $\delta\Delta(\textbf{r},t)$ relative to $\Delta_0$. These values are well within experimental feasibility and so we may neglect the self-consistent corrections to $\Delta(\textbf{r},t)$.

Since we are justfied in assuming $\Delta(\textbf{r},t)=\Delta_0$ the Green's function $\hat{\mathcal{G}}_0(x_1;x_2)$ appearing in Eq (\ref{eq:dyson}) is simply given by Eq (\ref{eq:rak}). By iterating Eq (\ref{eq:dyson}) it is straightforward to compute the full Green's function $\hat{\mathcal{G}}(x_1;x_2)$ in terms of the unperturbed Green's function $\hat{\mathcal{G}}_0(x_1;x_2)$ to aribitrary order in the drive $U_{\textbf{r},t}$. For $|U_{\textbf{r},t}|<<\mu$ we truncate the series to linear order in $U_{\textbf{r},t}$, and thus:
\begin{widetext}
\begin{equation}
\hat{\mathcal{G}}(\textbf{q},\omega;\textbf{Q},\Omega)\approx\delta(\textbf{Q})\delta(\Omega)\hat{\mathcal{G}}_0(\textbf{q};\omega) - \hat{\mathcal{G}}_0(\textbf{q}+\tfrac{\textbf{Q}}{2};\omega+\tfrac{\Omega}{2}) \hat{U}_{\textbf{Q},\Omega} \hat{\mathcal{G}}_0(\textbf{q}-\tfrac{\textbf{Q}}{2};\omega-\tfrac{\Omega}{2}) \\
\label{eq:dyson_linear}
\end{equation}
\end{widetext}
where $\textbf{q}$ and $\omega$ are the momentum and frequency conjugate to the relative position, $\textbf{r}_1-\textbf{r}_2$, and time, $t_1-t_2$, $\textbf{Q}$ and $\Omega$ are the momentum and frequency conjugate to the average position, $(\textbf{r}_1+\textbf{r}_2)/2$, and time, $(t_1+t_2)/2$, $\hat{U}_{\textbf{k},\omega}=\int d\textbf{r} dt e^{it\omega+i\textbf{r}\cdot\textbf{k}} \hat{U}_{\textbf{r},t}$, and $\hat{\mathcal{G}}_0(\textbf{q};\omega)$ is given by Eq (\ref{eq:rak}).

Focusing on the retarded components of $\hat{\mathcal{G}}(\textbf{q},\omega;\textbf{Q},\Omega)$ in Eq (\ref{eq:dyson_linear}) we find that the corrections to linear order in the drive are given by:
\begin{widetext}
\begin{equation}
\begin{aligned}
\delta\hat{G}^{R}(\textbf{q},\omega;\textbf{Q},\Omega) &= U_{\textbf{Q},\Omega}  \left[ \hat{G}^{R}_0(\textbf{q}+\tfrac{\textbf{Q}}{2};\omega+\tfrac{\Omega}{2})\hat{G}^{R}_0(\textbf{q}-\tfrac{\textbf{Q}}{2};\omega-\tfrac{\Omega}{2}) - \hat{F}^{R}_0(\textbf{q}+\tfrac{\textbf{Q}}{2};\omega+\tfrac{\Omega}{2})\hat{\overline{F}}^{R}_0(\textbf{q}-\tfrac{\textbf{Q}}{2};\omega-\tfrac{\Omega}{2}) \right] \\
\delta\hat{F}^{R}(\textbf{q},\omega;\textbf{Q},\Omega) &= U_{\textbf{Q},\Omega}  \left[ \hat{G}^{R}_0(\textbf{q}+\tfrac{\textbf{Q}}{2};\omega+\tfrac{\Omega}{2})\hat{F}^{R}_0(\textbf{q}-\tfrac{\textbf{Q}}{2};\omega-\tfrac{\Omega}{2}) - \hat{F}^{R}_0(\textbf{q}+\tfrac{\textbf{Q}}{2};\omega+\tfrac{\Omega}{2})\hat{\overline{G}}_0^{R}(\textbf{q}-\tfrac{\textbf{Q}}{2};\omega-\tfrac{\Omega}{2}) \right] \\
\delta\hat{\overline{G}}^{R}(\textbf{q},\omega;\textbf{Q},\Omega) &= U_{\textbf{Q},\Omega}  \left[ \hat{\overline{F}}^{R}_0(\textbf{q}+\tfrac{\textbf{Q}}{2};\omega+\tfrac{\Omega}{2})\hat{F}^{R}_0(\textbf{q}-\tfrac{\textbf{Q}}{2};\omega-\tfrac{\Omega}{2}) - \hat{\overline{G}}^{R}_0(\textbf{q}+\tfrac{\textbf{Q}}{2};\omega+\tfrac{\Omega}{2})\hat{\overline{G}}^{R}_0(\textbf{q}-\tfrac{\textbf{Q}}{2};\omega-\tfrac{\Omega}{2})\right] \\
\delta\hat{\overline{F}}^{R}(\textbf{q},\omega;\textbf{Q},\Omega) &= U_{\textbf{Q},\Omega}  \left[ \hat{\overline{F}}^{R}_0(\textbf{q}+\tfrac{\textbf{Q}}{2};\omega+\tfrac{\Omega}{2})\hat{G}_0^{R}(\textbf{q}-\tfrac{\textbf{Q}}{2};\omega-\tfrac{\Omega}{2}) - \hat{\overline{G}}^{R}_0(\textbf{q}+\tfrac{\textbf{Q}}{2};\omega+\tfrac{\Omega}{2})\hat{\overline{F}}^{R}_0(\textbf{q}-\tfrac{\textbf{Q}}{2};\omega-\tfrac{\Omega}{2}) \right].
\end{aligned}
\label{eq:r_linear}
\end{equation}
\end{widetext}

\subsection{Odd-frequency Terms in the Anomalous Green's Function}

Inserting the expressions from Eq (\ref{eq:rak}) to the equation for the linear corrections to the anomalous Green's function in Eq (\ref{eq:r_linear}) we find:
\begin{equation}
\delta\hat{F}^{R}(\textbf{q},\omega;\textbf{Q},\Omega) = i\hat{\sigma}_2   \dfrac{U_{\textbf{Q},\Omega} \Delta_0\left(\Omega + \xi_{\textbf{q}+\frac{\textbf{Q}}{2}} + \xi_{\textbf{q}-\frac{\textbf{Q}}{2}}\right) }{D_{\textbf{q},\omega;\textbf{Q},\Omega}D_{\textbf{q},\omega;-\textbf{Q},-\Omega}}
\label{eq:fr_linear}
\end{equation}
where
\begin{equation}
D_{\textbf{q},\omega;\textbf{Q},\Omega}\equiv(\omega+\tfrac{\Omega}{2})^2-\xi_{\textbf{q}+\frac{\textbf{Q}}{2}}^2-\Delta_0^2.
\label{eq:denomenator}
\end{equation}
By taking $\omega\rightarrow-\omega$ in Eq (\ref{eq:fr_linear}) we can see that this expression is neither even nor odd in $\omega$ and thus we have shown that the pair correlation function describing the driven condensate has both even- and odd- frequency terms.

It is straightforward to show that the odd-$\omega$ terms in $\delta\hat{F}^{R}(\textbf{q},\omega;\textbf{Q},\Omega)$ are given by:
\begin{equation}
\delta\hat{F}^{R}_{odd}(\textbf{q},\omega;\textbf{Q},\Omega) = i\hat{\sigma}_2 \ \mathcal{C}_{\textbf{q},\omega;\textbf{Q},\Omega} \ \omega\Omega\left( \xi_{\textbf{q}+\frac{\textbf{Q}}{2}} - \xi_{\textbf{q}-\frac{\textbf{Q}}{2}}\right)
\label{eq:odd_term}
\end{equation}
where we define the coefficient $\mathcal{C}_{\textbf{q},\omega;\textbf{Q},\Omega}= \frac{ U_{\textbf{Q},\Omega}\Delta_0\left(\Omega + \xi_{\textbf{q}+\frac{\textbf{Q}}{2}} + \xi_{\textbf{q}-\frac{\textbf{Q}}{2}}\right)}{D_{\textbf{q},\omega;\textbf{Q},\Omega}D_{\textbf{q},\omega;\textbf{Q},-\Omega}D_{\textbf{q},\omega;-\textbf{Q},\Omega} D_{\textbf{q},\omega;-\textbf{Q},-\Omega} } $ which is strictly even in $\omega$ and $\textbf{q}$.

Written in this form Eq (\ref{eq:odd_term}) clearly has singlet spin structure and is manifestly odd in $\omega$. Additionally, by taking $\textbf{q}\rightarrow-\textbf{q}$ and noting that $\xi_{-\textbf{q}}=\xi_{\textbf{q}}$ it is clear that this term is odd in parity. Hence, starting from a superconductor with only spin singlet even-parity even-$\omega$ ($\mathcal{S}=-1$, $\mathcal{T}=1$, $\mathcal{O}=1$, $\mathcal{P}=1$) pairs we have induced spin singlet odd-parity odd-$\omega$ ($\mathcal{S}=-1$, $\mathcal{T}=-1$, $\mathcal{O}=1$, $\mathcal{P}=-1$) pair amplitudes. We note that if $|\textbf{Q}|\rightarrow 0$ or $\Omega\rightarrow 0$ the ($\mathcal{S}=-1$, $\mathcal{T}=-1$, $\mathcal{O}=1$, $\mathcal{P}=-1$) amplitudes will vanish which is consistent with our expectation that the drive must possess both time-dependence and spatial inhomogeneity to transmute even-parity even-$\omega$ pairs into odd-parity odd-$\omega$ pairs.

Furthermore, we note that the odd-$\omega$ terms in $\delta\hat{F}^{R}(\textbf{q},\omega;\textbf{Q},\Omega)$ are a direct consequence of the denominator $D_{\textbf{q},\omega;\textbf{Q},\Omega}$ which acts to shift the spectrum in both frequency and momentum space. This denominator is shared by both $\delta\hat{F}^{R}(\textbf{q},\omega;\textbf{Q},\Omega)$ and $\delta\hat{G}^{R}(\textbf{q},\omega;\textbf{Q},\Omega)$ which is connected to observables. In section \ref{observables} we will explore measurable consequences of these shifted spectra.

\subsection{Time-Dependence of $\delta\hat{F}^{R}_{odd}$}

Since $\delta\hat{F}^{R}_{odd}(\textbf{q},\omega;\textbf{Q},\Omega)$ in Eq (\ref{eq:odd_term}) depends on the center-of-mass frequency ($\Omega$) and momentum ($\textbf{Q}$) we can deduce that it must have some nontrivial dependence on absolute time and position. We will now analyze this time- and space-dependence by transforming Eq (\ref{eq:odd_term}) to the mixed representation using the definition:
\begin{equation}
\hat{\mathcal{G}}(\textbf{q},\omega;\textbf{R},T)\equiv \int \frac{d\Omega}{2\pi}\frac{d\textbf{Q}}{(2\pi)^2} \hat{\mathcal{G}}(\textbf{q},\omega;\textbf{Q},\Omega) e^{-i\textbf{R}\cdot\textbf{Q}-iT\Omega}
\label{eq:mixed}
\end{equation}
together with a concrete expression for the drive:
\begin{equation}
U_{\textbf{r},t}=U_0\cos\left( \textbf{Q}_0\cdot\textbf{r} \right)\cos\left( \Omega_0 t \right).
\label{eq:drive}
\end{equation}

Using the expressions in Eqs (\ref{eq:odd_term})-(\ref{eq:drive}) we find that the odd-$\omega$ terms in $\delta\hat{F}^{R}(\textbf{q},\omega;\textbf{R},T)$ are given by:
\begin{widetext}
\begin{equation}
\delta\hat{F}^{R}_{odd}(\textbf{q},\omega;\textbf{R},T)= \omega \Delta_0i\hat{\sigma}_2 \frac{ U_0 \Omega_0  \sin(\textbf{Q}_0\cdot\textbf{R})  \left(  \xi_{\textbf{q}+\tfrac{\textbf{Q}_0}{2}} - \xi_{\textbf{q}-\tfrac{\textbf{Q}_0}{2}} \right) \left[ \sin(\Omega_0T)\left( \xi_{\textbf{q}+\tfrac{\textbf{Q}_0}{2}} + \xi_{\textbf{q}-\tfrac{\textbf{Q}_0}{2}} \right) + i\cos(\Omega_0T) \Omega_0 \right] }{D_{\textbf{q},\omega;\textbf{Q}_0,\Omega_0}D_{\textbf{q},\omega;\textbf{Q}_0,-\Omega_0}D_{\textbf{q},\omega;-\textbf{Q}_0,\Omega_0} D_{\textbf{q},\omega;-\textbf{Q}_0,-\Omega_0}} .
\label{eq:fr_mixed}
\end{equation}
\end{widetext}
Notice that $\delta\hat{F}^{R}_{odd}(\textbf{q},\omega;\textbf{R},T)$ possesses a finite real part which evolves in time, $T$, 90$^\circ$ out-of-phase with respect to the imaginary part of $\delta\hat{F}^{R}_{odd}(\textbf{q},\omega;\textbf{R},T)$ while both the real and imaginary parts possess the same spatial modulation. If we focus on the real part of Eq (\ref{eq:fr_mixed}) we see that it is exactly 90$^\circ$ out-of-phase with the drive and thus the odd-$\omega$ correlations are largest at the times and positions where the drive vanishes.

\section{Observable Features}
\label{observables}

We will now investigate the relationship between the dynamically-induced odd-$\omega$ pairing and two measureable quantities: the spectral function and density of states.

\subsection{Spectral Function}

In general, the spectral function can be related to the retarded components of the Green's function by:
\begin{equation}
A_{\textbf{k},\omega}(\textbf{R},T)=-\frac{1}{\pi}\text{Im}\text{Tr} \hat{\mathcal{G}}^{R}(\textbf{k},\omega;\textbf{R},T)
\label{eq:akw}
\end{equation}
where $\textbf{R}$ and $T$ are the average position and time. This quantity can be measured using photoemission spectroscopy\cite{damascelli2003angle}.

In the absence of an external drive, $A_{\textbf{k},\omega}(\textbf{R},T)$ for a mean-field superconductor is given by:
\begin{equation}
A^{(0)}_{\textbf{k},\omega}=2\left[ \delta\left( \omega - E_\textbf{k} \right) + \delta\left( \omega + E_\textbf{k} \right) \right]
\label{eq:a0}
\end{equation}
where $E_\textbf{k}=\sqrt{\xi_{\textbf{k}}^2 + \Delta_0^2}$. As we can see from Fig~\ref{fig:spec} (a), this spectral function exhibits two bands of quasiparticle states separated by a gap of $2\Delta_0$. These bands are each defined by the standard ``sombrero" shape with extrema appearing at $k=0$, $E_0=\pm\sqrt{\mu^2+\Delta_0^2}$ and $k=\sqrt{2m\mu}/\hbar$, $E_{k_F}=\pm\Delta_0$.

We will account for the presence of an external drive by using the expressions for the undriven Green's functions in Eq (\ref{eq:rak}) together with the linear order corrections to the retarded Green's functions in Eq (\ref{eq:r_linear}) to obtain the linear order corrections to $A_{\textbf{k},\omega}(\textbf{R},T)$:
\begin{widetext}
\begin{equation}
\begin{aligned}
A_{\textbf{k},\omega}(\textbf{R},T) &=A^{(0)}_{\textbf{k},\omega} + \cos(\textbf{R}\cdot\textbf{Q}_0+T\Omega_0)\delta A^{(1a)}_{\textbf{k},\omega} + \cos(\textbf{R}\cdot\textbf{Q}_0-T\Omega_0)\delta A^{(1b)}_{\textbf{k},\omega}, \\
\delta A^{(1a)}_{\textbf{k},\omega} &=\left[ f_{\textbf{k},\omega}(\textbf{Q}_0,\Omega_0) \delta\left( \omega + \frac{\Omega_0}{2} + E_{\textbf{k}+\tfrac{\textbf{Q}_0}{2}}\right) + f_{\textbf{k},\omega}(\textbf{Q}_0,-\Omega_0) \delta\left( \omega + \frac{\Omega_0}{2} - E_{\textbf{k}+\tfrac{\textbf{Q}_0}{2}}\right) \right. \\
&+\left. f_{\textbf{k},\omega}(-\textbf{Q}_0,\Omega_0) \delta\left( \omega - \frac{\Omega_0}{2} - E_{\textbf{k}-\tfrac{\textbf{Q}_0}{2}}\right) + f_{\textbf{k},\omega}(-\textbf{Q}_0,-\Omega_0) \delta\left( \omega - \frac{\Omega_0}{2} + E_{\textbf{k}-\tfrac{\textbf{Q}_0}{2}}\right) \right], \\
\delta A^{(1b)}_{\textbf{k},\omega} &=\left[ f_{\textbf{k},\omega}(\textbf{Q}_0,\Omega_0) \delta\left( \omega - \frac{\Omega_0}{2} - E_{\textbf{k}+\tfrac{\textbf{Q}_0}{2}}\right) + f_{\textbf{k},\omega}(\textbf{Q}_0,-\Omega_0) \delta\left( \omega - \frac{\Omega_0}{2} + E_{\textbf{k}+\tfrac{\textbf{Q}_0}{2}}\right) \right. \\
&+\left. f_{\textbf{k},\omega}(-\textbf{Q}_0,\Omega_0) \delta\left( \omega + \frac{\Omega_0}{2} + E_{\textbf{k}-\tfrac{\textbf{Q}_0}{2}}\right) + f_{\textbf{k},\omega}(-\textbf{Q}_0,-\Omega_0) \delta\left( \omega + \frac{\Omega_0}{2} - E_{\textbf{k}-\tfrac{\textbf{Q}_0}{2}}\right) \right]
\end{aligned}
\label{eq:akwrt}
\end{equation}
\end{widetext}
where we have assumed a drive of the form shown in Eq (\ref{eq:drive}) and, for convenience, we have defined the function:
$$f_{\textbf{k},\omega}(\textbf{Q},\Omega)=U_0\frac{\xi_{\textbf{k}-\tfrac{\textbf{Q}}{2}} + \xi_{\textbf{k}+\tfrac{\textbf{Q}}{2}}\left(1+\dfrac{\Omega}{E_{\textbf{k}+\tfrac{\textbf{Q}}{2}}}\right) }{E^2_{\textbf{k}-\tfrac{\textbf{Q}}{2}}-E^2_{\textbf{k}+\tfrac{\textbf{Q}}{2}}\left(1+\dfrac{\Omega}{E_{\textbf{k}+\tfrac{\textbf{Q}}{2}}}\right)^2}.$$

In Fig~\ref{fig:spec}(b) we plot Eq (\ref{eq:akwrt}) for $\textbf{R}=0$ and $T=0$ and find that the main effect of the driving potential is the generation of duplicate spectra shifted in momentum and frequency by $\pm\textbf{Q}_0/2$ and $\pm\Omega_0/2$ respectively. From the form of Eq (\ref{eq:akwrt}) this is clear because the delta functions apearing in $\delta A^{(1a)}_{\textbf{k},\omega}$ and $\delta A^{(1b)}_{\textbf{k},\omega}$ determine which values of $\textbf{k}$ and $\omega$ obtain nonzero spectral weight. Physically this means that the shifted spectral features are due to the driving field imparting energy $\pm\Omega_0/2$ and momentum $\pm\textbf{Q}_0/2$ to the undriven quasiparticles. These shifted spectral features are in fact just the first harmonics in an infinite series emerging from higher order powers in the drive.

It should be noted that these shifted spectral features can be traced back to the denominator from Eq (\ref{eq:denomenator}). Therefore, we can see a relation between the emergence of these harmonics and the odd-frequency terms in the anomalous Green's function. However, it is only for drives which possess nonzero $\textbf{Q}_0$ and $\Omega_0$, like the case shown in Fig~\ref{fig:spec} (b), that the generation of duplicate spectra coincides with the emergence of odd-$\omega$ terms in the anomalous Green's function.

In Fig.~\ref{fig:spec} (c) we show $A_{\textbf{k},\omega}(\textbf{R},T)$ for the case of finite driving frequency and zero wavevector, $\Omega_0\neq0$ and $\textbf{Q}_0=0$, which corresponds to the case of driving the superconductor with a light source. In Fig.~\ref{fig:spec} (d) we plot the opposite limit of zero driving frequency and finite wavevector, $\Omega_0=0$ and $\textbf{Q}_0\neq0$, which corresponds to subjecting the superconductor to a static spatially nonuniform electric field. Notice that in both cases additional spectra are generated as expected from Eq (\ref{eq:akwrt}).

Turning our attention to the case of $\textbf{R},T\neq0$, we can see from Eq (\ref{eq:akwrt}) that $A_{\textbf{k},\omega}(\textbf{R},T)$ possess periodic modulation in both $\textbf{R}$ and $T$. In the special case where $\textbf{R}\cdot\textbf{Q}_0+T\Omega_0=\textbf{R}\cdot\textbf{Q}_0-T\Omega_0=(2n+1)\pi/2$ we can see that the linear order corrections to the spectral function vanish and only the undriven spectrum remains. However, for all other values of $\textbf{R}$ and $T$ the additional spectra remain. Furthermore, in the case where $\textbf{R}\cdot\textbf{Q}_0+T\Omega_0=\textbf{R}\cdot\textbf{Q}_0-T\Omega_0=2n\pi$ we get exactly the results appearing in Fig~\ref{fig:spec}. For other values of $\textbf{R}$ and $T$ the relative weight and signs of the different shifted bands will change; however, the delta functions in Eq (\ref{eq:akwrt}) ensure that these features will be located at the same values of $\textbf{k}$ and $\omega$ as the features shown in Fig~\ref{fig:spec}.

\begin{figure}
 \begin{center}
  \centering
  \includegraphics[width=0.5\textwidth]{spectral_4panel.eps}
  \caption{(color online) Comparison of the spectral function, $A_{\textbf{k},\omega}(\textbf{R},T)$, for both an undriven and driven 2D superconductor described by the Hamiltonian in Eq.~(\ref{eq:Ham}). All energies are presented in units of the undriven order parameter, $\Delta_0$, all wavevectors are presented in units of the inverse coherence length $\xi^{-1}=\pi\Delta_0/\hbar v_F$ and plotted along the $k_x$-direction. (a) Spectral function for an undriven $s$-wave superconductor using Eq (\ref{eq:a0}). Horizontal lines at $\pm E_{0}=\pm\sqrt{\mu^2 + \Delta_0^2}$ denote the energies associated with notable extrema of the ``sombrero" shape. (b), (c), \& (d) Spectral functions for a driven $s$-wave superconductor calculated using Eq (\ref{eq:akwrt}) for $\textbf{R}=0$, $T=0$, with $U=0.25$, but for different pairs of $\Omega_0$ and $\textbf{Q}_0$: (b) $\Omega_0=0.25$, $\textbf{Q}_0=(0.25,0)$; (c) $\Omega_0=0.25$, $\textbf{Q}_0=0$; and (d) $\Omega_0=0$, $\textbf{Q}_0=(0.25,0)$. In each case vertical and horizontal lines denote the shifted spectra arising at $\pm\textbf{Q}_0/2$ and $\pm\Omega_0/2$.
         }
  \label{fig:spec}
 \end{center}
\end{figure}

\subsection{Density of States}

Another important observable quantity is the electronic local density of states (LDOS), which can be calculated from the spectral function using
\begin{equation}
\mathcal{N}(\omega,\textbf{R},T)= \int \frac{d\textbf{q}}{(2\pi)^2} A_{\textbf{q},\omega}(\textbf{R},T)
\label{eq:dos}
\end{equation}
and can be measured by scanning tunneling microscopy\cite{hofer2003theories,tersoff1983theory}.

In Fig.~\ref{fig:dos} (a) we plot the LDOS for an undriven superconductor using Eqs (\ref{eq:a0}) and (\ref{eq:dos}). We can see that the undriven LDOS exhibits the trends of an $s$-wave superconductor, two coherence peaks separated by a gap of $2\Delta_0$. Notice the two shoulders at $\pm E_0$ just beyond the coherence peaks associated with the extrema at the center of the ``sombrero" shape in the spectra.

In Fig.~\ref{fig:dos} (b) we show the driven LDOS at $\textbf{R}=0$, $T=0$, calculated using Eqs (\ref{eq:akwrt}) and (\ref{eq:dos}), for a single driving frequency of $\Omega_0=\Delta_0/4$ and overlay the plots for three different values of the driving wavevector $|\textbf{Q}_0|=Q_0$. The plots for different wavevectors have been shifted to make a clearer comparison, in each case the LDOS vanishes at the center of the gap. Two notable features emerge in the driven LDOS: (i) pairs of satelite peaks appear around the coherence peaks at $\pm\Delta_0\pm\Omega_0/2$; and (ii) pairs of singular points appear at $E_0\pm\Omega_0/2$. By comparing the features at different values of $Q_0$ we can see that the additional features at $E_0\pm\Omega_0/2$ hold for all $Q_0$ while the additional peak structures near the coherence peaks possess a strong dependence on $Q_0$.

We note that these satelite coherence peaks arise only for nonzero $Q_0$ and $\Omega_0$, identically to the odd-$\omega$ terms in the anomalous Green's function. Furthermore, by comparing the plots in Figs~\ref{fig:spec} and \ref{fig:dos} we can see that the origin of these satelite coherence peaks appears to be from the positive spectral features shifted by $\textbf{Q}_0/2$ to the outside of the original spectra and then up and down in energy by $\Omega_0/2$. While we expect the satelite features near $E_0$ to appear in any driven superconductor regardless of the wavevector, the peaks appearing at $\Delta_0\pm\Omega_0/2$ seem to emerge only within the parameter regime in which the odd-$\omega$ terms are expected to develop in the pairing amplitude. This connection makes sense when we consider the fact that both the shifted spectral features and the odd-$\omega$ pairing terms have a common origin in the denominator of the retarded Green's function. Therefore, the shifted peaks in this system offer a possible means to infer the presence of the odd-$\omega$ pair amplitudes.

\begin{figure}
 \begin{center}
  \centering
  \includegraphics[width=0.5\textwidth]{dos_2panel.eps}
  \caption{(color online) LDOS, $\mathcal{N}(\omega,\textbf{R}=0,T=0)$, for both the case of an undriven and driven 2D superconductor described by the Hamiltonian in Eq.~(\ref{eq:Ham}) with a driving potential of the form $U_{\textbf{r},t}=U\cos(\textbf{r}\cdot\textbf{Q}_0)\cos(t\Omega_0)$. (a) LDOS for undriven $s$-wave superconductor with vertical lines denoting the shoulders associated with the energies marked in Fig~\ref{fig:spec}(a). (b) LDOS for driven $s$-wave superconductor with $U=0.25$ and $\Omega_0=0.25$ evaluated for three different driving wavevectors $\textbf{Q}_0=(Q_0,0)$. Vertical lines mark the positions of: shoulder at $E_{0}$ (solid grey) and satelite features appearing at $E_0\pm\Omega_0/2$(dashed-dot grey), along with the satelite features associated with the coherence peak at $\Delta_0\pm\Omega_0/2$ (dashed black). Driven LDOS plots separated for clarity, in each case the LDOS vanishes within the gap.
         }
  \label{fig:dos}
 \end{center}
\end{figure}

\section{Conclusions}

Motivated by the intrinsically dynamical nature of odd-frequency pairing we have generalized Berezinskii's classification of superconducting pairing symmetries to driven systems. Subsequently, we argued that under certain conditions odd-frequency pairing should arise as a consequence of a time-dependent driving potential and investigated this effect explicitly for a particular model of a conventional $s$-wave superconductor coupled to a time-dependent and spatially nonuniform electric field. Computing the corrections to the superconducting correlation functions in this system using linear response we found dynamically-induced odd-frequency terms appearing in the anomalous Green's function. Based on these results we can conclude that in order to generate odd-frequency pair amplitudes at this order using a spin-independent drive the external potential must be both time-dependent and spatially nonuniform. Furthermore, we demonstrated that these odd-frequency terms are related to features appearing in both the density of states and the spectral function, offering an experimental signature of their presence.

Acknowledgements: We wish to thank Annica Black-Schaffer, Matthias Eschrig, Mats Horsdal, Yaron Kedem, Sergey Pershoguba, and Enrico Rossi for useful discussions. This work was supported by US DOE BES E304 VR (AVB) and the European Research Council (ERC) DM-321031 (CT).

redpajama_set_name: RedPajamaArXiv


text:
Q: Split a Pandas DataFrame column based on a pattern I have a pandas DataFrame which looks something like this
d1 = pd.DataFrame({"Title":['t1','t2','t3'],
               "Description":['[rank:102] [status: Confirmed] [City: xyz] [Country:C1]',
                              '[rank:93] [status: Rejected] [City: abc] [Country:C2]',
                              '[rank: ] [status: Confirmed] [City: ] [Country: ]']})

As you can see, the important data which is needed for analysis is in the second column in a pattern which I want to separate.
What is the best way to separate 'rank','status','city' and 'country into different columns themselves.
Also, sometimes the data will will be absent in the description column for a few rows (record 3 for example)
As my final output, I would want my dataset to contain five columns

*

*Title

*Rank

*Status

*City

*Country

A: You could use a regex with str.extractall to extract the key/value pairs as two columns, then pivot to set the first column as column names, and finally join to the original:
(d1
 .drop('Description', axis=1)
 .join(
  d1['Description']
  .str.extractall('([^][]+):([^][]+)')
  .droplevel(1) # get rid of extra "match" level to aggregate the rows into 1
  .pivot(columns=0, values=1)
 )
)

output:
  Title  City Country rank      status
0    t1   xyz      C1  102   Confirmed
1    t2   abc      C2   93    Rejected
2    t3                      Confirmed

redpajama_set_name: RedPajamaStackExchange


I am a loyal, dedicated, hard working and responsible person. I work in the advertising industry and enjoy traveling, watching sports, going to the gym, bike riding, and eating at different restaurants. I'd love to find a companion to share this journey of life with.
I'm looking for a woman with a good, kind heart. Someone who is affectionate, considerate, faithful, and sincere. Someone who wants to settle down with someone and be in a committed relationship.
Not a lot, only a few real friends.
She's kind, faithful, polite, considerate, appreciative, honest, loving, and has integrity.


text:
// Copyright (c) Dyoub Applications. All rights reserved.
// Licensed under MIT (https://github.com/dyoub/app/blob/master/LICENSE).

(function () {

    function Controller(handleError, path, PurchasePayments) {
        this.handleError = handleError;
        this.path = path;
        this.PurchasePayments = PurchasePayments;
    }

    Controller.prototype.addPayment = function () {
        var controller = this,
            totalPayable = controller.purchaseOrder.totalPayable,
            totalPaid = controller.purchaseOrder.totalPaid,
            paymentList = controller.purchaseOrder.paymentList;

        paymentList.push({
            date: new Date(),
            total: (totalPayable - totalPaid).round(2)
        });

        controller.calculateTotalPaid();
    };

    Controller.prototype.calculatePayment = function (payment) {
        var controller = this;
        controller.calculateTotalPaid();
    };

    Controller.prototype.calculateTotalPaid = function () {
        var controller = this;
        controller.purchaseOrder.totalPaid = 0;

        if (controller.purchaseOrder.paymentList) {
            controller.purchaseOrder.paymentList.forEach(function (payment) {
                controller.purchaseOrder.totalPaid += (payment.total || 0);
            });
        }
    };

    Controller.prototype.calculateTotalPayable = function () {
        var controller = this,
            discount = controller.purchaseOrder.discount || 0,
            shippingCost = controller.purchaseOrder.shippingCost || 0,
            otherTaxes = controller.purchaseOrder.otherTaxes || 0,
            total = controller.purchaseOrder.total,
            totalPayable = total + shippingCost + otherTaxes
                - (controller.purchaseOrder.total * discount / 100);

        controller.purchaseOrder.totalPayable = totalPayable.round(2);
    };

    Controller.prototype.hideRemoveDialog = function () {
        var controller = this;
        controller.paymentToRemove = null;
    };

    Controller.prototype.init = function () {
        var controller = this;

        controller.routeParams = controller.path
            .map('/purchase-orders/edit/:purchaseOrderId/items');

        controller.searchPayments();
    };

    Controller.prototype.noPayments = function () {
        var controller = this;
        return !controller.searchingPayments &&
            controller.purchaseOrder.paymentList &&
            controller.purchaseOrder.paymentList.isEmpty();
    };

    Controller.prototype.paidOut = function () {
        var controller = this;
        return controller.purchaseOrder &&
            controller.purchaseOrder.totalPaid === controller.purchaseOrder.totalPayable;
    };

    Controller.prototype.removePayment = function () {
        var controller = this,
            payment = controller.paymentToRemove,
            paymentIndex = controller.purchaseOrder.paymentList.indexOf(payment);

        controller.purchaseOrder.totalPaid -= payment.total;
        controller.purchaseOrder.paymentList.splice(paymentIndex, 1);
        controller.paymentToRemove = null;
    };

    Controller.prototype.purchaseHasProducts = function () {
        var controller = this;
        return controller.purchaseOrder &&
            controller.purchaseOrder.total > 0;
    };

    Controller.prototype.save = function () {
        var controller = this;

        if (controller.saveBlocked() || controller.saving) {
            return;
        }

        controller.saving = true;

        controller.PurchasePayments.save({
            purchaseOrderId: controller.routeParams.purchaseOrderId,
            shippingCost: controller.purchaseOrder.shippingCost,
            otherTaxes: controller.purchaseOrder.otherTaxes,
            discount: controller.purchaseOrder.discount,
            payments: controller.purchaseOrder.paymentList
        })
        .then(function () {
            controller.path.redirectTo('/purchase-orders/details/:purchaseOrderId',
                controller.routeParams);
        })
        ['catch'](function (response) {
            controller.handleError(response);
            controller.saving = false;
        });
    };

    Controller.prototype.saveBlocked = function () {
        var controller = this;
        return controller.saving ||
            controller.paymentsForm.$invalid ||
            !controller.paidOut();
    };

    Controller.prototype.searchPayments = function () {
        var controller = this;
        controller.searchingPayments = true;

        controller.PurchasePayments
            .list(controller.routeParams.purchaseOrderId)
            .then(function (response) {
                controller.purchaseOrder = response.data;
                controller.notFound = !controller.purchaseOrder;

                if (controller.purchaseOrder) {
                    controller.calculateTotalPayable();
                    controller.calculateTotalPaid();
                }
            })
            ['catch'](function (response) {
                controller.handleError(response);
            })
            ['finally'](function () {
                controller.searchingPayments = false;
            });
    };

    Controller.prototype.searchPaymentsComplete = function () {
        var controller = this;
        return !(controller.searchingPayments || controller.error);
    };

    Controller.prototype.showRemovePaymentDialog = function (payment) {
        var controller = this;
        controller.paymentToRemove = payment;
    };

    angular.module('dyoub.app').controller('PurchasePaymentsEditController', [
        'HandleError',
        'Path',
        'PurchasePayments',
        Controller
    ]);

})();

redpajama_set_name: RedPajamaGithub


text:
Lunch Money Sells Out (Again)
Yet another printing of Lunch Money is now history. The reprint has already been ordered; delivery is expected in early November. This is a straight reprint -- no changes. Lots of our distributors recently stocked up on the game, upon hearing that we were running low, so your favorite game store probably still has it in stock (and if not, their distributor probably does) -- if you're a consumer, this out-of-stock period is probably not even going to be noticeable. But if you need a copy for a specific date/purpose (like a birthday present), I wouldn't put off buying it.
Ars Magica Reprints in October
Based on several factors (the success of the True Lineages reprint, the short time in which Mystery Cults sold out, the generally good sales of the whole 5th Edition line, and the fact that the printing cost looks affordable), we've decided to arrange a modest reprinting of two out of print Ars Magica 5th Edition supplements: Guardians of the Forests and the ENnie Award-nominated Houses of Hermes: Mystery Cults. These reprints will be available in October. (Specifically, our printer has advised me that October 17th is the expected delivery date.)
Guardians of the Forests
Stock No. AG0276, ISBN 1-58978-072-8, MSRP $29.95
Reprint Ships to Distributors in October 2006
Houses of Hermes: Mystery Cults
In case you are wondering -- no, these are not print-on-demand; we are doing a regular print run on web/offset presses, just like the first printings of these books. These are reprints, not revisions (stock number, price, ISBNs remain the same), so people who already own the books won't want to buy them again; however, we believe there is unsatisfied demand for these books among the game's fans which these reprintings will meet.
City & Guild Slipping to November
Jury duty recently pulled Jeff Tidball away from his layout work on City & Guild. Unfortunately, we don't think it's possible for this book to be out on time in October; we're expecting November delivery instead, and hoping that the above-mentioned reprints will hold the Ars Magica fans at bay in the meantime.
Well, if Motherland calls, little to do. Best wishes to Jeff for performing his duty as best as it can. Is this delay going to affect Ancient Magic release date, as well ?
John Nephew 9/12/2006 8:53 AM
No, Ancient Magic looks right on time. It should have a release date in very early January.

redpajama_set_name: RedPajamaCommonCrawl


text:
Diamond Guitars > Overview guitars > Products > Gibson Electrics > Gibson L-5 Wes Montgomery (SOLD)
Gibson L-5 Wes Montgomery (SOLD)
Category: Gibson Electrics
Our price:€0,00
Here's a well taken care of Gibson L-5 Custom Wes Montgomery built in 1997 by the Master Luthiers of the Gibson Custom Shop. Having been in continues production since 1922, the L-5 is the pinnacle of the Gibson archtop collection, in no small part to the fact that Wes Montgomery made it his guitar of choice during his short, but highly successful career.
Other players that have chosen the L-5 as their axe of choice are Tuck Andress, Lee Ritenour, Pat Martino, and Eddie Lang. Though these guitars come with a hefty price tag, the L-5 is still the Gibson guitar that many players save their pennies for as they search for the perfect (Wes Montgomery inspired) tone. The Gibson L-5 is the Holy Grail of archtops, the kind of guitar that has it all: looks, tone, playability and the patina of a history extending back to 1922 and through the hands of an amazing array of players that speaks to its versatility. Also John Mayer, Eric Clapton, Keith Richards, Django Reinhardt, Paul Simon and Scotty Moore are among the musical royalty who have used an L-5 in the studio or on stage.
Wes Montgomery is the most impartant jazz guitar player of the century. And that is an understatement! Wes had such a profound influence on any one musical genre, much less one as conspicuously diverse as jazz music. Yet in the span of just a few years in the late 1950s and early 1960s, jazz legend Wes Montgomery was able forge a body of work that is still referred to today as the most important and influential of any music ever recorded by a jazz artist. Montgomery's mastery of the guitar is well documented, and the enormous impact on his peers is recognized time and time again, from Pat Metheny to George Benson to Joe Satriani, all of whom claim Montgomery as the virtuoso who inspired them to reach new heights.
Wes ordered various L-5 version and in finally Gibson built his own master model: the L-5 Custom Wes Montgomery. The Custom Artist Model L-5 has a single "Classic 57" pickup in the neck position and an X-brace supporting the top. These features Montgomery felt helped achieve the warm guitar voice for which he loved this guitar model.
We bought this guitar from the first owner whe treated the instrument very well. She has a lot of playing hours but was never on stage. Out luthier gave her the full cleaning and polishing treatment and a fret leveling job. This guitar is in Excellent Plus condition
Solid Carved Sitka Spruce top
Solid flamed Maple back and sides
Multi-ply black and white binding
Single-ply white binding around the f-holes
Venetian cutaway
Gold ABR-1 bridge with Ebony base
L-5 tailpiece with 'Wes Montgomery' inscription
25½-inch scale length neck
Five-piece flamed Maple neck
20-fret Ebony fingerboard
Pearl block inlays
Multi-ply black and white body bindings
ES-rounded neck profileGibson '57 Classic PAF pickup
Flower pot inlay on the headstock
Schaller M6 tuners
Vintage Sunburst
Gibson USA Hard shell case
Gibson EH-185 Lapsteel Guitar (1939)
Gibson Explorer Sammy Hagar Red Rocker Signature (SOLD)
Gibson Byrdland Master Collection Jim Triggs with Venetian Cutaway (1991)
Gibson L-4 CES Custom Shop – Wine Red (2010)


If you were strolling in Paris this last Saturday, and many other French cities (as well as in Belgium), you might have come across crowds of people wearing kraft signs with various tags on it: « I vote for my planet », « planet in danger », « make my planet great for once », « 49§3 pour le climat ».
We were over 100 000 people worldwide marching the streets for climate this past Saturday, in many cities in France and all over the world.
800 gatherings took place that day! A major event that comes after a few years of catastrophic ecologic policy in the US under president Donald Trump, a year of Don Quijote policy under Ecology Minister Nicolas Hulot, whom resigned in late August. His resignation gave birth to these marches. Its motive is to put climate in the heart of political debates. We were there!
The Climate Marches can be seen as a sign that people finally realize how severe the situation is.
In his resignation speech, ecology minister Nicolas Hulot stroke the French population by explaining how little politics can do without civilian mobilization.
He complained that he didn't have structured back-ups to take to the streets along with him, he blame the country for not being united for our future and our children's. This Saturday, we demonstrated the opposite: we DO care about climate. Despite our governments running on an economic model that not only disrespects nature but is destroying it, despite powerful lobbies, despite eco-sceptics leaders, let us show how strong we can be and how much we can achieve together.
September, a Climate Walk – October a Climate Tsunami!
Let not this march be a one-time thing. Let ecology and climate remain a main topic to be taken care of by governments. We can't do it without them, but they won't do it without a constant pressure FROM US!
Gather at 11.00am for an assembly where we'll discuss action plans, talk about collective demands.
And this in as many cities in the world as possible! Climate is the business of the Citizens of the World, not just a single country or people.
We need you, because we need our planet. JOIN US!
For a bigger impact, you can join United4Earth (not to be mistaken with unitedforearth.com, with whom we are going to merge forces with) in order to group together and organise internationally!
Our main target is to get our citizen Lobbying tool up and running so that it can help us all pressure elected officials in order to pass laws to protect our environment.
With many, many, many other organisation and citizen initiative we are trying to get as many cities walking as possible on October 13th 2018!
Rise for Climate – Decorate our windows to protest!
This entry was posted in Climat Actions. Bookmark the permalink.


text:
Boutique-y unveils 52yo single grain
Jul 07, 2016 by Annie Hayes
Whisky bottler That Boutique-y Whisky Company (TBWC) has released a 52-year-old single grain Scotch whisky in collaboration with London's Black Rock Whisky Bar.
The independent bottler's latest launch is from the now-closed Carsebridge distillery, located in Scotland's Lowlands, and features a voucher on the back of the bottle entitling the buyer to a secret off-menu cocktail at Black Rock. Known for their graphic-novel style labels, TBWC's latest release features the bar's founders, Tristan Stephenson and Thomas Aske, sat at the Black Rock's 18-foot European oak tree trunk table – depicting the two 17-litre channels used for ageing whiskies and cocktails, as well as the taps used for serving them. Best known as the entrepreneurs behind Fluid Movement bar consultants and Worship Street Whistling Shop, the duo opened Black Rock – which carries more than 250 different whiskies from around the world – in Shoreditch back in March. A statement from TBWC reads: "Their mission is to demystify whisky and make it simple and fun for all, something which is also fundamental to everything TBWC do." The expression retails at the RRP of £231.95, and is available from That Boutique-y Whisky Company retailers including Amazon, Master of Malt, Selfridges, Hedonism Wines, Majestic Wines and Harvey Nichols.
Boutique-y unveils 52yo single grain - Wooden Cork


VIT (Deemed to be University) was founded in 1984 as a self financing institution called the Vellore Engineering College. It is now a Deemed to be University. It also has campuses in Chennai (Tamil Nadu), Bhopal (Madhya Pradesh) and Amravati (Andhra Pradesh).
(a) Toppers of each State Board and Central Board : 100% Tuition fee waiver for all the four years.
(b) VITEEE rank holders of 1 to 50 : 75% Tuition fee waiver for all the four years.
(c) VITEEE rank holders of 51 to 100 : 50% Tuition fee waiver for all the four years.
(d) VITEEE rank holders of 101 to 1000 : 25% Tuition fee waiver for all the four years.
Counselling : The selected candidates will have to submit a DD for Rs.50,000 /- in favour of 'Vellore Institute of Technology", payable at Vellore, on the confirmation of seat and acceptance of branch allotted. This includes `11,800/- towards counselling process fee. Counselling process fee is NON-REFUNDABLE, but will be adjusted in the tuition fee once the candidate joins the institute..
Mock Test : A mock test will be available to the candidates on the institute website. This is mainly designed to provide the candidates awareness of the procedure and to get used to the various components of the exam. The mock test will give the candidate an idea of all the features of the entrance test. The mock test will be in the form of flash file that can be downloaded to the system and the candidate can go through the entire flash file of the demo/mock test any number of times.


This is one of a set of four paintings of domestic servants and craftsmen that we have for sale (see stock codes JM-658, JM-660, JM-664, JM-666).
There is paint loss, cracking and marks to the mica at the upper right, and to the lower edge, as shown. The colour denoting the ground is partially painted on the back of the mica, where there are two small patches of paint loss. Other lines and marks visible in the photo are not cracks but rather lines of slight delamination, which are less prominent in reality. The mica is loose, not laid down on any backing.


retirement investment with plans that offer growth and protect your future income.
A fixed annuity is a long-term, tax-deferred* insurance contract or investment designed for retirement. It allows you to create a fixed stream of income during your retirement through a process called annuitization.
An investor can purchase an annuity with a lump sum or monthly payments. The money is invested for a future date, when the money can be withdrawn by the investor either monthly, quarterly, annually or in a lump sum. That money can also be guaranteed for a specific period or for a lifetime.
If a person has an annuity with a guaranteed 10-year income, and that person dies after five years, the beneficiary will receive payments for the remaining five years.
If a person has an annuity with lifetime payments, beneficiaries receive nothing upon the death of the holder of the annuity. It is possible to purchase an annuity with lifetime payments and a set number of guaranteed years. In that case, if the person dies before the end of the guaranteed years, beneficiaries would receive the remaining guaranteed funds.
Money invested in an annuity can grow tax-deferred*. When the money is withdrawn, only the investment earnings are taxed, not the amount of money invested.
Similar to a 401(k) or other defined contribution plans, money from an annuity can be subject to an additional 10 percent tax penalty – on top of your current tax rate – if it is withdrawn before age 59 ½.
*Tax-Deferred: Tax-deferred means taxes on your interest earnings are postponed until a future point in time. By postponing taxes, your money compounds faster because you earn interest on dollars that would have otherwise been paid to the IRS. This enables you to accumulate more money over a shorter period of time, ultimately providing you with greater income.


text:
4 New Views of Essential Tremor
Focused Ultrasound
Diagnosing ET
Deep Brain Stimulation - DBS
Essential tremor (ET) is the most common movement disorder, with at least 200,000 new U.S. cases annually. While ET is not life-threatening, research has led to four new lines of thinking about ET.
1. A family of diseases
Doctors and researchers are backing off from a universal approach to ET. Differences in ET patterns exist, hinting that ET is not a single disorder. Some patients develop walking/balance difficulties that make falls a hazard, while others have tremors localized only to hands or neck. There are subgroups of patients whose eye movements become abnormal, or who develop cognitive dysfunction. Recent studies on deceased patients' brains suggest the ET doesn't always affect brains the same way. Thus, there may ET subcategories – or even distinct diseases. This may help explain why a drug works with one patient but not another. Work is ongoing to organize ET into diagnostic categories that could lead to more effective treatment matching.
2. Focusing on the cerebellum
A small area at the back of the brain is called the cerebellum (Latin for little brain). It is responsible for coordinating voluntary movement, posture, balance, and motor control. Cerebellum activity has "outflow pathways" that lead to the thalamus deep in the brain's center. When the cerebellum or its outflow pathways are not functioning properly, the thalamus receives incorrect signals that it relays to the muscles, causing tremors. Surgical procedures (deep brain stimulation, thalamotomy) act upon the thalamus to block these erroneous signals from reaching the muscles.
3. ET as a neurodegenerative disorder
Once ET begins, it has a gradual progression toward worsening tremors and possible problems in other body areas. Cognitive/emotional difficulties may show up. As people age, severity can increase to a point where ET has a profound impact on independence and quality of life. ET is a process that takes a cumulative toll over time. Research is aimed at identifying risk factors that trigger cell death or the buildup of harmful proteins, in the hope of creating preventive therapies.
4. The role of GABA
Researchers have identified a particular chemical messenger called GABA that helps control motor movements by acting as a pacemaker. One aspect of ET as a neurodegenerative disorder appears to be the dying off of cells in the cerebellum that manufacture GABA. As GABA levels drop off, cerebellum activity goes into high gear, resulting in "hyperactive" messages to the thalamus.
Together, these four views of ET represent the most recent evolution in how doctors and scientists understand ET. The promise of a better life with ET, or eliminating it altogether, lies with them.
About Dr. Dan Sperling
Dan Sperling, MD, DABR, is a board certified radiologist who is globally recognized as a leader in multiparametric MRI for the detection and diagnosis of a range of disease conditions. As Medical Director of the Sperling Prostate Center, Sperling Medical Group and Sperling Neurosurgery Associates, he and his team are on the leading edge of significant change in medical practice. He is the co-author of the new patient book Redefining Prostate Cancer, and is a contributing author on over 25 published studies. For more information, contact the Sperling Neurosurgery Associates.
Success! Please be sure to add neuro@sperlingmedical.com to your contacts to ensure you always receive our articles and news.
Meet Our Neurosurgeon
https://youtu.be/CS5xBEFvjBQ
Other Treatments (5)
Cognitive Health (3)
Essential Tremor Research (2)
Diagnosing ET (3)
Wellness and Essential Tremor (2)
Research & Studies (3)
Brain Disorders (2)
Exercise & Health (1)
Deep Brain Stimulation – DBS (9)
Brain Science (7)
Focused Ultrasound (25)
Quality of Life (15)
Nutrition/Wellness (11)
Essential Tremor (77)
Resources & Support (3)
Alternative Therapies (2)