text:
\subsection{\label{sec:A} Introduction}

Far-from-equilibrium systems beyond the paradigm of linear response has been a subject of growing theoretical and experimental interest \cite{sskj,qh1,qh2,qh3,qh4,qh5,KLH2015,ah0}. Such studies are important from fundamental perspective and from the point of view of device applications. Especially, understanding the role of interactions is of great interest. Important fundamental problems to explore include, role of far-from-equilibrium physics in physical quantities of interest and the intricate interplay between system and bath degrees of freedom when interactions are involved. Some of the main consequences of interactions range from fundamental physics (such as the phenomenon of Kondo effect \cite{kontosPRL,keijiPRL}) to device applications (such as diodes/rectifiers\cite{kul2015,spins_diode1,spins_diode2}). Even though the role of interaction in device applications in fermionic systems is well studied \cite{Dubi2015,Chang2010,Ono2002}, the corresponding bosonic set-ups are much less explored. However, with the advent of the relatively new field of photonics, bosonic set-ups for device applications such as a potential optical diode are gaining popularity \cite{Wang2013, Shen2014, Mascarenhas2014}.  With recent cutting-edge technology in hybrid quantum systems, it is now possible to design interacting bosonic Hamiltonians and reservoirs \cite{ber1,ah0,ah1,ah2,ah3,ah4,ah5}.  There is progress not only in fabricating Hamiltonian-Reservoir systems, but also, great advance in measuring physical quantities of interest such as photon number, photon statistics \cite{petta0} and photon current \cite{Google2016} .

One of the most basic interacting bosonic Hamiltonian system that one can think of is a single bosonic site with Bose-Hubbard  interaction, hereafter, called the single-site Bose Hubbard (SSBH) model. In optics, such interaction is often called the Kerr interaction strength. In the field of hybrid quantum systems a Hamiltonian with such an interaction can be potentially experimentally realized in more than one way. The Jaynes-Cummings model, an experimentally realized light-matter system, can be tuned to the dispersive regime where it behaves like a SSBH model. The photon-spin interaction in this limit can be integrated out ``pertubatively'' to generate a non-linear Bose-Hubbard like interaction between photons \cite{Blais2009,ah1}. Alternatively there are interesting potential realizations of Bose-Hubbard interactions between photons which involve 4-level atoms in an optical cavity \cite{Hartman2007,Ling2011}. These realizations offer large tunability of parameters. Specifically, while the former realization involving Jaynes-Cummings model has a small interaction strength compared to the linear term, the later realizations have very large interaction strength.

Another area of applicability for such interacting bosonic Hamiltonians, which is perhaps more suited to non-equilibrium measurements, are the fields of molecular thermoelectrics and nanophononics. These fields typically deal with molecular junctions connecting two reservoirs, experimentally, which are often two large chemical compounds \cite{exp1,exp2,exp3,exp4,exp5,exp6}. Role of interactions in phononic transport through such systems is of interest both experimentally and theoretically\cite{ds1}.

There has been a large amount of work on such a SSBH model with a finite interaction strength coupled to a single bath \cite{Walls1980,Haake1986,Milburn1986,Haake1987,Alicki1989,Kartner1993}. On the other hand, non-equilibrium spin boson (NESB) model, which corresponds to the limit of very large interaction strength, has been well studied in out-of-equilibrium set-ups \cite{SegalPRL2005, SegalJChemPhys2005, SegalPRB2006, NESBcond1, keijiPRL, NESBcond2} and is of growing experimental significance. The conductance of an anharmonic junction with quartic anharmonicity has been studied recently \cite{jt1,jt2}.
However, there is essentially no investigation of the SSBH model with finite interaction strength in a far-from-equilibrium setup via connection with multiple reservoirs. In this paper, we investigate the SSBH model weakly coupled to two bosonic baths at different temperatures beyond the linear response regime.

The organization of our paper is as follows: In section~\ref{sec:B}, we describe our setup. In section~\ref{sec:C}, we give the Redfield quantum master equation for our setup and in section~\ref{sec:D}, we find the non-equilibrium steady state (NESS) properties.  In section~\ref{sec:E}, we look at scaling behaviour of average particle number and energy at NESS and in section~\ref{sec:F}, we discuss the scaling behaviour and rectification of NESS particle and energy currents.  In section ~\ref{sec:TD}, we present results for time dynamics of various physical quantitites which has become of growing recent interest. Finally, in section~\ref{sec:G}, we summarize our main results along with an outlook.

\subsection{\label{sec:B} Model and Setup}
We consider a single site with Bose-Hubbard interaction connected to two bosonic baths in non-equilibrium. The baths are taken to be quadratic and the system-bath couplings are taken to be bilinear. Thus our set-up is given by the full system+bath Hamiltonian
\begin{align}
&\hat{H} = \hat{H}_S+\hat{H}_B+\hat{H}_{SB} \nonumber\\
&\hat{H}_S = \Omega_0 \hat{a}^\dagger\hat{a} + \chi (\hat{a}^\dagger\hat{a})^2 \nonumber\\
&\hat{H}_B = \sum_{\ell=1}^2\sum_{r=1}^\infty \Omega_r^\ell \hat{B}_r^{\ell \dagger} \hat{B}_r^\ell~, \\
&\hat{H}_{SB} = \varepsilon\sum_{\ell=1}^2 \sum_{r} (\kappa_{\ell r} \hat{B}_r^{\ell \dagger} \hat{a} + \kappa_{\ell r}^* \hat{a}^{\dagger} \hat{B}_r^\ell)~,\nonumber
\end{align}
where $\hat{a}$ correspond to bosonic annihilation operators and  $\hat{B}_{r}^{\ell}$ to those of $\ell$th bath ($\ell=\{1,2\}$).  Note that  $\hat{N}=\hat{a}^\dagger\hat{a}$ is a conserved quantity with respect to the system Hamiltonian $H_S$. The energy spectrum of the system Hamiltonian can be easily written down and it has the non-linear form $\hat{E}=\Omega_0 \hat{N} + \chi \hat{N^2}$.

The baths are quadratic and have infinite degrees of freedom. $\varepsilon$ is a dimensionless parameter that controls system bath coupling. We assume that, initially, there is no coupling between the system and the baths, and the two baths are at thermal equilibrium with their own inverse temperature $\beta_1$ and $\beta_2$ and chemical potential $\mu_1$ and $\mu_2$. Thus the  initial bath correlation functions satisfy the thermal properties :
\begin{equation}
\label{initial_bath_corr}
\begin{split}
\langle\hat{B}_r^\ell\rangle = 0 , \hspace{2pt} <\hat{B}_r^{\ell\dagger} \hat{B}_s^\ell >_B = \mathfrak{n}_{\ell}(\Omega_r^\ell)\delta_{r s }~,
\end{split}
\end{equation}
where $\mathfrak{n}_\ell(\omega)= [{e^{\beta_{\ell}( \omega-\mu_{\ell})}- 1}]^{-1}$ is the bosonic distribution function. We also introduce the bath spectral functions:
\begin{equation}
\mathcal{J}_{\ell}(\omega)=2\pi \sum_r  \mid \kappa_{\ell r} \mid^2 \delta(\omega - \Omega_r^\ell)~.
\end{equation}

We will assume,
\begin{align}
\mathcal{J}_\ell(\omega) = \Gamma_\ell \mathcal{J}(\omega)~,
\end{align}
i.e, the two baths have same density of states, but the system-bath coupling is different in general. $\Gamma_\ell$ has dimensions such that $\mathcal{J}_\ell(\omega)$ has dimensions of energy.

The quantitative nature of the results will depend on our choice of spectral function. We assume a general spectral function which is commonly used in bosonic systems:
\begin{align}
\label{J_general}
\mathcal{J}(\omega)=\omega^s e^{-\omega/\omega_c}~,
\end{align}
where $\omega_c$ gives the cut-off frequency. The cut-off frequency is considered very large so that the system energy levels near the edge of the bath spectrum correspond to extremely high energies, which do not really contribute to the system properties at the chosen set of temperatures and chemical potentials. This is satisfied when $\omega_c \gg \Omega_0, \chi, \beta_1^{-1}, \beta_2^{-1},\mu_1,\mu_2$. Also, we are concerned with a photonic or phononic system, so we will set $\mu_1=\mu_2=0$ finally.

\subsection{\label{sec:C} The Quantum Master Equation (QME)}

We want to investigate this set-up without any restrictions on the interaction strength $\chi$. So, we adopt the method of QME under Born-Markov approximation, which is only valid when system-bath coupling is weak, and gives results only to leading order in system-bath coupling. The microscopically derived Redfield QME for our system coupled to single bath was first written down in \cite{Haake1986}. It is straighforward to generalize to two baths. Let us define $\hat{\rho} \equiv Tr_B(\hat{\rho}_{full})$ with  $\hat{\rho}_{full}$ being the full density matrix of system+bath and $Tr_B(..)$ implying trace taken over bath degrees of freedom. The  weak-coupling Redfield equation for the density matrix of the system $\hat{\rho}$ of our set-up is given by :

\begin{align}
\label{rqme}
&\frac{\partial \hat{\rho}}{\partial t} = i[\hat{\rho},\hat{H}_S] - \varepsilon^2 \Big ( [\hat{\rho} F(\hat{\tilde{\omega}})\hat{a},\hat{a}^\dagger]  + [\hat{a}^\dagger, G(\hat{\tilde{\omega}})\hat{a} \hat{\rho}] + h.c. \Big)~,
\end{align}
where
\begin{align}
\label{FG}
&\hat{\tilde{\omega}} = \Omega_0+\chi(2\hat{N}+1), \nonumber \\
&F(\hat{\tilde{\omega}}) = F_1(\hat{\tilde{\omega}})+F_2(\hat{\tilde{\omega}}), \hspace{5pt} G(\hat{\tilde{\omega}}) = G_1(\hat{\tilde{\omega}})+G_2(\hat{\tilde{\omega}})  \\
& F_\ell(\hat{\tilde{\omega}}) = \frac{1}{2} \mathcal{J}_\ell(\hat{\tilde{\omega}})\mathfrak{n}_\ell(\hat{\tilde{\omega}}) - i \mathcal{P}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi} \frac{\mathcal{J}_\ell(\omega)\mathfrak{n}_\ell(\omega)}{\omega-\hat{\tilde{\omega}}} \nonumber\\
& G_\ell(\hat{\tilde{\omega}}) = \frac{1}{2} \mathcal{J}_\ell(\hat{\tilde{\omega}})(\mathfrak{n}_\ell(\hat{\tilde{\omega}})+1) - i \mathcal{P}\int_{-\infty}^{\infty} \frac{d\omega}{2\pi}\frac{\mathcal{J}_\ell(\omega)(\mathfrak{n}_\ell(\omega)+1)}{\omega-\hat{\tilde{\omega}}} \nonumber
\end{align}
and $h.c.$ stands for Hermitian conjugate. Note that the Redfield equation assumes $\varepsilon \ll 1$ and keeps terms only upto $O(\varepsilon^2)$, i.e, only upto quadratic in system-bath coupling (the Born approximation). In deriving the above equation, we have also done the Markov approximation which assumes that the observation time is much larger than the time scale of relaxation of bath correlation functions.   To ensure weak system-bath coupling, we choose $\varepsilon=0.1$ and $\kappa_{\ell r}$ to be of the same order as $\Omega_0$.

The Markov assumption entails that Eq.~\ref{rqme} is not valid at short times. In fact, as pointed out in Ref \cite{Alicki1989}, this equation is not completely positive and may lead to unphysical states at small times for certain initial conditions. A way around suggested in Ref \cite{Alicki1989} is to derive Lindblad equation in terms of eigenbasis operators of $\hat{H}_S$ via the secular approximation. The equation so derived respects complete positivity at all times. In the following, we will be interested only in the diagonal elements of $\hat{\rho}$ in the eigenbasis of $\hat{H}_S$. It turns out that the evolution equation for these elements as derived from the Redfield equation and from the eigenbasis Lindblad equation are exactly the same.

In equilibrium, i.e, when $\beta_1=\beta_2=\beta$, $\mu_1=\mu_2=\mu$, it can be checked by direct substitution that the thermal state
\begin{align}
\label{rho_eq}
\hat{\rho}_{eq} = \frac{e^{-\beta (\hat{H}_S - \mu \hat{N}})}{Z}
\end{align}
is the steady state of the  Eq.~(\ref{rqme}), where $Z~=~Tr\big [e^{-\beta (\hat{H}_S - \mu \hat{N}}\big]$ is the equilibrium partition function.

\subsection{\label{sec:D} The Non-equilibrium Steady State (NESS)}

To calculate various physical observables in NESS beyond linear response,  we need to find the NESS density matrix. Also, in non-equilibrium, since we have no guess for direct substitution, we need to find the $\rho$ directly from Eq \ref{rqme}. Since both the number operator $\hat{N}$ and the system Hamiltonian $H_S$ are diagonal in occupation number basis, the NESS transport properties as well as average occupation and energy of the system  can be found from the steady state diagonal elements of $\hat{\rho}$ in this basis. The occupation number basis satisfies
\begin{align}
\hat{a}|n\rangle = \sqrt{n} |n-1\rangle, \hspace{5pt} \hat{a}^\dagger|n\rangle = \sqrt{n+1} |n+1\rangle~.
\end{align}
The  evolution equation for the diagonal elements $\rho_n = \langle n | \hat{\rho} | n \rangle$ is given by
\begin{align}
\label{rho_n_equation}
\frac{d \rho_n}{dt} = -\varepsilon^2 \left[ \rho_n ( C_n + D_n)- \rho_{n-1}C_{n-1} - \rho_{n+1}D_{n+1} \right]~,
\end{align}
where
\begin{align}
\label{CD}
&C_n = C^{(1)}_{n}+C^{(2)}_{n}, \hspace{7pt} D_n = D^{(1)}_{n}+D^{(2)}_{n} \nonumber \\
&C^{(\ell)}_{n} = (n+1)\mathcal{J}_\ell(\omega_n)\mathfrak{n}_\ell(\omega_n), \nonumber \\
&D^{(\ell)}_{n} = n\mathcal{J}_\ell(\omega_{n-1})(\mathfrak{n}_\ell(\omega_{n-1})+1) \nonumber \\
&\omega_n = \Omega_0+\chi(2n+1)~.
\end{align}
In the steady state we set the LHS of Eq.~\ref{rho_n_equation}) to zero. This leads to a difference equation. Noting  that $C^{(\ell)}_{-1}=C_{-1} = 0, D^{(\ell)}_{0}=D_0 = 0$,
 we obtain,  by recursion, the solution
\begin{align}
&\rho_n D_n = \rho_{n-1}C_{n-1} \label{rho_n_condition}\\
&\Rightarrow\rho_n = \rho_0\prod_{p=1}^{n}\frac{C_{p-1}}{D_p} =\rho_0\prod_{p=1}^{n} \frac{\sum_{\ell=1}^2\Gamma_\ell \mathfrak{n}_\ell(\omega_{p-1})}{\sum_{\ell=1}^2\Gamma_\ell ( \mathfrak{n}_\ell(\omega_{p-1})+1)} \nonumber \\
&\textrm{for} \hspace{5pt} n={1,2,3,...}  \label{rho_n}~.
\end{align}
The constant $\rho_0$ is fixed from the normalization condition $\sum_{n} \rho_n=~1$, i.e, trace of density matrix is unity. Thus,
\begin{align}
\label{Z_def}
\rho_0 = \tilde{Z}^{-1} = \Big[1+\sum_{n=1}^{\infty}  \prod_{p=1}^{n} \frac{\sum_{\ell=1}^2\Gamma_\ell \mathfrak{n}_\ell(\omega_{p-1})}{\sum_{\ell=1}^2\Gamma_\ell ( \mathfrak{n}_\ell(\omega_{p-1})+1)}   \Big]^{-1}~,
\end{align}
where $\tilde{Z}$ is a normalization constant and is analogous to the partition function in equilibrium systems. It can be easily checked by putting $\mathfrak{n}_1(\omega)=\mathfrak{n}_2(\omega)=\mathfrak{n}(\omega)$ in Eq.~(\ref{rho_n}) that $\rho_n$ indeed gives Eq.~(\ref{rho_eq}) in equilibrium. Also, in the equilibrium case, Eq.~(\ref{rho_n_condition}) corresponds to the detailed balance condition. In the non-equilibrium case, while this still looks like the detailed balance condition, we note that in general it is not possible to define an effective temperature.
Also note that $\rho_n$ is independent of the choice of spectral function of the bath.

\begin{figure}[t]
\includegraphics[scale=0.45]{rho_n.pdf}
\caption{(color online) The plot of population density (diagonal elements of density matrix in eigenbasis of system Hamiltonian) of SSBH model in equilibrium and under thermal bias with assymetric system-bath coupling ($\Gamma_1=0.4,\Gamma_2=1.6$) and $\Omega_0=1$. The top and bottom panels are for interchanged hot and cold baths. The three dotted lines (in each plot) are the corresponding equilibrium distributions ($T_1=T_2=3.5$), i.e, Eq. \ref{rho_eq} for the values of $\chi$ mentioned in the legend. The deviation from equilibrium is more prominent in the bottom panel. For large interaction strength ($\chi=4$), only two levels have non-negligible probability, like a spin-boson model. All energy variables are measured in units of $\Omega_0$.} \label{fig:one_rho_n}
\end{figure}

The explicit expression for population $\rho_n$ of bosons in NESS, given by Eq.~\ref{rho_n}, is the central result that allows us to go beyond linear response in this interacting bosonic problem. The population, in itself, is a physically measurable quantity and, as we show below, can be used to compute various other physical observables.

In Fig.~\ref{fig:one_rho_n}, we show the plots of population density $\rho_n$ of the system under asymmetric system-bath coupling. Since system-bath coupling is asymmetric, the out-of-equilibrium distribution changes under interchange of hot and cold baths. For high interaction strength, i.e, for $\chi \gg \Omega_0, T_1,T_2$, only the lowest two levels have non-negligible probability. In this regime, we can truncate the energy spectrum in just two levels. Then $\rho_0$ and $\rho_1$ become
\begin{align}
\label{rho_SB}
\rho_0 \approx \frac{\sum_{\ell=1}^2 \Gamma_\ell \big(\mathfrak{n}_\ell(\omega_0) +1 \big) }{\sum_{\ell=1}^2 \Gamma_\ell (1+2\mathfrak{n}_\ell(\omega_0))},& \hspace{2pt}
\rho_1 \approx \frac{\sum_{\ell=1}^2 \Gamma_\ell \mathfrak{n}_\ell(\omega_0) }{\sum_{\ell=1}^2 \Gamma_\ell (1+2\mathfrak{n}_\ell(\omega_0))} \nonumber \\
& \hspace{10pt} \forall \hspace{5pt} \chi \gg \Omega_0, T_1,T_2
\end{align}
with $\omega_0 = \Omega_0+\chi$. The above results are exactly the same as obtained for a non-equilibrium spin-boson model (NESB) by using the Redfield equation \cite{SegalPRL2005}.
Thus for $\chi \gg \Omega_0, T_1, T_2$, the system becomes identical to the NESB.
Since NESB is already a well explored problem, in the following, we will be mainly interested in the physics beyond this regime.
We also note that for $\chi=0$ the system reduces to a harmonic oscillator and in this case the population $\rho_n^{\chi=0}$ is given in terms of an effective temperature $T_{eff}=1/\beta_{eff}$, i.e  $\rho_n^{\chi=0} \propto e^{-\beta_{eff} \Omega_0 n}$ with
\begin{align}
\coth (\beta_{eff}\Omega_0)= \frac{\Gamma_1 \coth (\beta_{1}\Omega_0)+\Gamma_2 \coth (\beta_2\Omega_0)}{\Gamma_1+\Gamma_2}~ \label{beta_eff}
\end{align}
which is consistent with the  finding in Ref. \cite{dhar2012}.

\subsection{\label{sec:E} Average occupation and energy}

First we will look at the average occupation and energy of the system. These quantities are measurable in current state-of-the-art experiments in quantum light-matter hybrid systems \cite{ah1,petta0}. The expressions for these are given by
\begin{align}
\label{occ_and_E}
&\langle \hat{N} \rangle  = \sum_{n=1}^{\infty} n \rho_n = \frac{1}{\tilde{Z}}\sum_{n=1}^{\infty} n \prod_{p=1}^{n} \frac{\sum_{\ell=1}^2\Gamma_\ell \mathfrak{n}_\ell(\omega_{p-1})}{\sum_{\ell=1}^2\Gamma_\ell ( \mathfrak{n}_\ell(\omega_{p-1})+1)}~, \nonumber\\
&\langle \hat{H}_S \rangle  = \sum_{n=1}^{\infty} E_n \rho_n = \frac{1}{\tilde{Z}}\sum_{n=1}^{\infty} E_n \prod_{p=1}^{n} \frac{\sum_{\ell=1}^2\Gamma_\ell \mathfrak{n}_\ell(\omega_{p-1})}{\sum_{\ell=1}^2\Gamma_\ell ( \mathfrak{n}_\ell(\omega_{p-1})+1)}~,
\end{align}
with $E_n = \Omega_0 n+\chi n^2$.

In NESB limit, these average quantities can be trivially found from Eq.~\ref{rho_SB}. They become $\langle \hat{N} \rangle \approx \rho_1$, and $\langle \hat{H}_S \rangle \approx \omega_0 \rho_1$. We are interested in going beyond the NESB regime. So let us look at the regime of high temperatures $T_1, T_2 \gg \chi,\Omega_0$. First, we look at the normalization constant defined in Eq.~\ref{Z_def}. which can be written as
\begin{align}
&\tilde{Z} = 1+ \sum_{n=1}^\infty \exp\left[-\sum_{p=1}^{n} \log\left(f(\omega_{p-1}) \right)\right] \nonumber \\
&f(\omega_p)=  \frac{\sum_{\ell=1}^2\Gamma_\ell ( \mathfrak{n}_\ell(\omega_{p})+1)}{\sum_{\ell=1}^2\Gamma_\ell \mathfrak{n}_\ell(\omega_{p})} ~.
\end{align}
Note that  $f(\omega_p)>1$, and hence $log\big(f(\omega_p)\big)>0$. It follows that there is a energy level cut-off $n^*$ beyond which the energy levels have negligible contribution to $\tilde{Z}$. For high enough temperatures, we can assume, $\beta_\ell \omega_{p} \ll 1$ $\forall p<n^*$. Under this condition, we can expand $f(\omega_p)$ to obtain (after some amount of algebra),
\begin{align}
 f(\omega_p) &\approx 1+\frac{\omega_p}{\tilde{T}}, \nonumber \\
{\rm hence}~~log\big( f(\omega_p) \big) &\approx log \big(1+\frac{\omega_p}{\tilde{T}}\big) \approx \frac{\omega_p}{\tilde{T}}~,
\end{align}
with
\begin{align}
\label{Teff}
\tilde{T} = \frac{\Gamma_1 T_1 + \Gamma_2 T_2}{\Gamma_1 + \Gamma_2}~.
\end{align}
The contribution of terms $n> n^*$ is small and so their precise form is irrelevant. Hence we get
\begin{align}
\tilde{Z}&\approx 1+\sum_{n=1}^{\infty} \exp\left[-\frac{1}{\tilde{T}}\sum_{p=1}^{n} \big( \Omega_0 + (2p-1)\chi \big) \right]
\nonumber \\
&=\sum_{n=0}^{\infty} \exp\left[-\frac{1}{\tilde{T}}( \Omega_0 n +\chi n^2 )\right]~.
\end{align}
Thus, for $T_1,T_2\gg \Omega_0,\chi$ the normalization constant has the same form as the equilibrium partition function with the effective temperature $\tilde{T}$. This is consistent with the effective temperature for harmonic oscillator ($\chi=0$) given in Eq.~\ref{beta_eff}. For high temperatures, $\beta_{eff}={1}/{\tilde{T}}$. It is also interesting to note that for symmetric system-bath coupling, i.e, $\Gamma_1=\Gamma_2$, the effective temperature is just the mean temperature, $\tilde{T}=T_m=(T_1+T_2)/2$. However, the description in terms of an effective temperature is not possible for low temperatures, except when $\chi = 0$.

The high temperature scaling of the normalization constant can now be easily found by noting that for high temperatures, the summation can be converted into an integral. So we have
\begin{align}
\tilde{Z}\approx \int_0^\infty dx \exp\left[-\frac{1}{\tilde{T}}( \Omega_0 x +\chi x^2 )\right] \approx \frac{\sqrt{\pi}}{2}\sqrt{\frac{\tilde{T}}{\chi}}~.
\end{align}
The second step requires the condition $\tilde{T}\chi \gg \Omega_0^2$, under which  we see that  the normalization constant scales as $\sqrt{\tilde{T}/\chi}$.

\begin{figure}[t]
\includegraphics[scale=0.42]{occ_and_energy_scaling.pdf}
\caption{(color online) The plot shows scaling behaviour of average occupation and average energy of the single site Bose-Hubbard model for fixed $r={T_2}/{T_1}={1}/{3}$. Here $\omega_0 = \Omega_0+\chi$. The dotted plots correspond to the NESB model. For $\chi \gg \Omega_0$ ($\Omega_0=1$), there is data collapse over the entire range of temperature. Even for small $\chi (=0.1)$, there is substantial deviation from linear ($\chi=0$) behaviour. For high temperatures, $\langle \hat{N} \rangle$ scales as $\sqrt{T_1/\omega_0}$, while, $\langle \hat{H}_S \rangle$ scales as $T_1$. Here the system-bath coupling is taken symmetric : $\Gamma_1=\Gamma_2=1$. All energy variables are measured in units of $\Omega_0$.} \label{fig:occ_and_energy}
\end{figure}

The above trick can be used to find high temperature scaling of average of any operator which is diagonal in the eigenbasis of the system Hamiltonian. The average of any operator $\hat{h}$ which is diagonal in the eigenbasis of the system has the form
\begin{align}
\label{avg_form}
\langle \hat{h}\rangle = \sum_{n=0}^\infty \rho_n h(n)=\frac{[h(0)+\sum_{n=1}^\infty e^{-\sum_{p=1}^{n} \log\left(f(\omega_{p-1}) \right)}h(n)]}{\tilde{Z}}~,
\end{align}
which, for $T_1,T_2\gg \Omega_0,\chi$,  exactly following above arguments, becomes
\begin{align}
\label{avg_form_highT}
\langle \hat{h}\rangle \approx \frac{\sqrt{2}}{\pi}\sqrt{\frac{\chi}{\tilde{T}}} \int_0^\infty dx \exp{\left[-\frac{ \Omega_0 x +\chi x^2 } {\tilde{T}}\right]} h(x)~.
\end{align}
Using this, we readily obtain the high temperature behaviour of  $\langle \hat{N} \rangle$ and $\langle \hat{H}_S \rangle$
\begin{align}
\label{occ_energy_highT}
\langle \hat{N} \rangle \approx \sqrt{\frac{\tilde{T}}{\pi \chi}}, \hspace{3pt}
\langle \hat{H}_S \rangle \approx \tilde{T}+ (\Omega_0+2\chi) \sqrt{\frac{\tilde{T}}{\pi \chi}}~.
\end{align}
Thus $\langle \hat{N} \rangle$ should show a data collapse for various $\chi$ and vary as a function of $\sqrt{{\tilde{T}}/{\chi}}$, whereas, $\langle \hat{H}_S \rangle/\chi$ varies as ${\tilde{T}}/{\chi}$ and should show a data collapse for $\chi \gg \Omega_0$.

To check the above high temperature discussion and the connection with equilibrium behaviour, we now first define
\begin{align}
r=\frac{T_2}{T_1}~,
\end{align}
which quantifies the degree of deviation from equilibrium, $r~=~1$ corresponding to  equilibrium. To ensure that the system is far from equilibrium, we keep $r$ fixed at $r<1$.  We note that, if $r$ is kept fixed, all the NESS results become a function of only one temperature, say, $T_1$. We choose $T_1/\omega_0$, where $\omega_0=\Omega_0+\chi$,  as the scaling variable since it can be used both in the highly interacting regime (where $\chi \approx \omega_0$) and the linear regime (where $\chi=0$). Also in the NESB regime, with $r$ fixed, all NESS quantities vary as a function of $T_1/\omega_0$, as can be checked from Eq.~\ref{rho_SB}.

In Fig.~\ref{fig:occ_and_energy}, we plot $\langle \hat{N} \rangle$ and $\langle \hat{H}_S \rangle/\omega_0$ as a function of $T_1/\omega_0$. Note that Eq.~\ref{occ_and_E}, and not any simplified expression, was used to calculate $\langle \hat{N} \rangle$ and $\langle \hat{H}_S \rangle$ in the plots. The plots clearly show data collapse over the entire temperature regime for $\chi \gg \Omega_0$. Also, important to note, is the substantial effect of small interaction strengths at high temperatures. The high temperature scaling behaviour for small interaction strengths is same as that for large interaction strengths, but there is no data collapse. This is because, the condition $\tilde{T}\chi \gg \Omega_0^2$ means that, for small $\chi$, there are large subleading terms. The low temperature behaviour matches with NESB. Since all these observations are valid for any choice of $r$, it follows that they are all valid in equilibrium (i.e, $r=1$) also. This is consistent with previous results in \cite{Haake1986}.

The crucial point in above discussion was to find that high temperature non-equilibrium results can be described via an effective temperature $\tilde{T}$. In the following, we will see that even transport properties at high temperatures can be described in terms of  $\tilde{T}$.

\subsection{\label{sec:F}Currents}

\begin{figure}[t]
\includegraphics[scale=0.45]{currents.pdf}
\caption{(color online) Plot of particle (top panel) and energy (bottom panel) currents as a function of interaction strength $\chi$ for transport under both forward and backward bias for Ohmic baths ($s=1$ in Eq.~\ref{J_general}) under asymmetric system bath coupling ($\Gamma_1 =0.4,\Gamma_2=1.6$). Here mean temperature $T_m=(T_1+T_2)/2=5$ for both plots.  The two black dotted lines in each plot correspond to the NESB currents for $\Delta T = \pm 5$. The currents from our model match the NESB currents for large $\chi$. Particle current decreases with $\chi$, while energy current behaves non-monotonically with $\chi$. The forward and backward currents do not match, thereby showing rectification effect. The direction of rectification of energy current is reversed beyond a value of $\chi$. It follows that at this value of $\chi$, the energy current does not show rectification. The insets show the corresponding currents as a function of $\Delta T= T_1-T_2$ for a chosen value of $\chi=3$ and $T_m=5$. It can be seen that $I$ and $J$ deviate from odd-function behaviour which is signature of rectification.
Other parameters are $\varepsilon=0.1$, $\omega_c=1000$. All energy variables are measured in units of $\Omega_0$, and time is measured in units of $\Omega_0^{-1}$.  }
\label{fig:two_current}
\end{figure}

\begin{figure*}[t]
\includegraphics[scale=0.5]{currents_scaling.pdf}
\caption{(color online) Panel (a) and (b) show the scaling behaviour of $I/\Delta T$ and $J/(\Delta T\omega_0)$ for Ohmic baths ($s=1$ in Eq.~\ref{J_general}) and for fixed $r=\frac{T_2}{T_1}=\frac{1}{3}$. The dotted lines show the corresponding NESB result. The horizontal dashed lines in panel (a) and panel (b) correspond to $I/\Delta T = J/(\Delta T\omega_0)=A={\varepsilon^2 \Gamma_1 \Gamma_2}/{(\Gamma_1+\Gamma_2)}$, which is the high temperature result for the harmonic oscillator ($\chi=0$).  For $\chi \gg \Omega_0$ ($\Omega_0=1$), there is a data collapse for all temperatures.  The NESB result matches with the non-equilibrium SSBH model for small temperatures. At higher temperatures, the NESB result shows non-monotonicity, which is not seen in the non-equilibrium SSBH model which demonstrates a stark difference between the two models. In panel (a), $I/\Delta T$ approaches a constant value at high temperatures irrespective of the strength of interaction strength. In panel (b), even for small interaction strength ($\chi=0.1$), substantial deviation from linear ($\chi=0$) behaviour is noticeable.  Panel (c) shows data collapse of $J/(I\omega_0)$ for all temperatures for $\chi \gg \Omega_0$. Irrespective of choice of bath spectral function and for fixed $r(=\frac{1}{3})$, $J/(I\omega_0)$ shows a data collapse and goes as $\sim \sqrt{T_1/\omega_0}$ for $T_1\gg \omega_0$. All observations are also valid in linear response regime (i.e, $r\approx 1$), and give the temperature scaling of conductance.  Other parameters are:  $\varepsilon=0.1$, $\omega_c=1000$, $\Gamma_1=\Gamma_2=1$. All energy variables are measured in units of $\Omega_0$, and time is measured in units of $\Omega_0^{-1}$.  } \label{fig:currents}
\end{figure*}

Now we look at the average transport properties of the system, in particular we compute particle and energy currents.
To calculate current we look at the evolution equations of the expectation values $\langle \hat{N} \rangle$ and $\langle \hat{H}_S \rangle$ of $\hat{N}$ and $H_S$. Since $\hat{N}$ and $ \hat{H}_S$ are diagonal in the eigenbasis of the system Hamiltonoan we can directly obtain the evolution of their expectation values from Eq.~(\ref{rho_n_equation}). This gives
\begin{align}
\frac{d\langle \hat{N} \rangle}{dt} =& \sum_n n \frac{d\rho_n}{dt} =
 \varepsilon^2 \sum_n \Big (  \rho_n C_n - \rho_n D_n \Big)~, \label{N_eqn}\\
\frac{d\langle \hat{H}_S \rangle}{dt} = & \sum_n (\Omega_0 n +\chi n^2) \frac{d\rho_n}{dt} \nonumber \\ &= \varepsilon^2 \sum_n \Big ( \omega_n \rho_n C_n - \omega_{n-1} \rho_n D_n \Big)~.
\end{align}
Collecting all terms depending on each bath separately, the above equations can be written like continuity equations of the forms $d\langle \hat{N}\rangle/dt=  I_{1} - I_{2},
d\langle \hat{H}_S \rangle/dt=  J_{1} - J_{2}$,
where $I_{\ell}$ ($J_{\ell}$) is the particle (energy) current flowing into the system from $\ell$th bath. In steady state, $I_{1}=I_{2}=I$ and $J_{1}=J_{2}=J$. The steady state expressions for currents are :
\begin{align}
I& = \varepsilon^2\sum_{n=0}^{\infty} \rho_n (C^{(1)}_{n}-D^{(1)}_{n}) =\sum_{n=1}^{\infty} \rho_n n \mathcal{I}(\omega_{n-1})~,\nonumber \\
J& = \varepsilon^2\sum_{n=0}^{\infty} \rho_n (\omega_n C^{(1)}_{n}-\omega_{n-1} D^{(1)}_{n}) = \sum_{n=1}^{\infty} \rho_n \omega_{n-1} n \mathcal{I}(\omega_{n-1})~, \label{currents}
\end{align}
with
\begin{align}
\label{curr_op}
\mathcal{I}(\omega_{n-1}) = \varepsilon^2\Big [ \frac{\Gamma_1\Gamma_2\mathcal{J}(\omega_{n-1})(\mathfrak{n}_1(\omega_{n-1})-\mathfrak{n}_2(\omega_{n-1})) }{\Gamma_1\mathfrak{n}_1(\omega_{n-1})+\Gamma_2\mathfrak{n}_2(\omega_{n-1})}   \Big ]~.
\end{align}
The second steps of Eq.~\ref{currents} have been arrived at from the first steps after some simplification using the property of NESS density matrix given in Eq.~\ref{rho_n_condition}.
Note that energy and particle currents are not independent. But, in general, there is no way of directly finding one current given the other and they can have quite different behaviour. Fig.~\ref{fig:two_current} shows variation of energy and particle currents with interaction strength $\chi$ for Ohmic baths (($s=1$ in Eq.~\ref{J_general})), for both forward ($\Delta T >0$) and backward ($\Delta T <0$) biases. The mean temperature $T_m=(T_1+T_2)/2$ is kept fixed in the plots, and system-bath coupling is asymmetric ($\Gamma_1 \neq \Gamma_2$). The plots immediately show us a number of physical aspects of the system.

Firstly, we note that the particle current decreases with increase in interaction strength $\chi$. This is expected because of increasing repulsive interaction in the system. On the other hand, energy current shows non-monotonic behaviour with $\chi$. This is plausible because, while, with increasing $\chi$, system allows less number of particles to pass, higher energy particles have a larger probability to pass through the system.

Secondly, we see that there is rectification of both energy and particle currents, since the particle and energy currents for forward and backward biases do not match. This is to be expected because the expressions for currents in Eq.~\ref{currents} are not antisymmetric under interchange of hot and cold baths (i.e, $\mathfrak{n}_1\leftrightarrow \mathfrak{n}_2$) in general. It is only so under special conditions. Two of such special conditions where there is no rectification are when $\chi=0$, i.e, when the system is linear, and when $\Gamma_1=\Gamma_2$ (for any $\chi$). These can be easily checked from the expressions for currents (Eq.~\ref{currents}).  Hence, in general, there will be rectification effects in both particle and energy currents for $\chi\neq 0$ and $\Gamma_1 \neq \Gamma_2$.
This is the generic behaviour in non-linear (interacting) systems.

Thirdly, as discussed before, for $\chi \gg \Omega_0, T_1, T_2$, the system behaves as NESB, and currents match with the NESB results. But, the rectification in the NESB limit is less than that for smaller interaction strengths. Thus rectification behaviour is non-monotonic as a function of $\chi$. Our findings therefore suggest that a careful engineering of the system Hamiltonian is required to get maximum rectification from a given system.

Finally, and most interestingly, for small interaction strength, the rectification of energy current occurs in the opposite direction to rectification of particle current. Also, there is a non-zero value of $\chi$ where the forward and backward energy currents match, and hence there is no rectification. At this point, the system rectifies particle current but not the energy current. Beyond this value of $\chi$, energy and particle rectification occur in the same direction (see Fig. \ref{fig:two_current}, bottom panel).\\

In what follows, we investigate the behaviour of particle and energy currents and rectification in more detail along with their scaling behaviour.

\subsubsection{Scaling behaviour of currents}
As we have seen with average system properties, transport properties also behave differently for different relative values of temperatures and interaction strength. In the NESB regime, $\chi \gg \Omega_0, T_1, T_2$, the currents are given by
\begin{align}
\label{currents_SB}
&I_{SB} \approx   \frac{\varepsilon^2\Gamma_1\Gamma_2\mathcal{J}(\omega_{0})(\mathfrak{n}_1(\omega_{0})-\mathfrak{n}_2(\omega_{0})) } {\Gamma_1 (1+2\mathfrak{n}_1(\omega_0))+\Gamma_2 (1+2\mathfrak{n}_2(\omega_0))}~, \nonumber\\
&J_{SB} \approx \omega_0  I_{SB}~,
\end{align}
with $\omega_0 = \Omega_0+\chi$. This is identical to the expression for current previously derived for NESB \cite{SegalPRL2005}. Note that in NESB regime, energy current is proportional to particle current. This is because, in this limit, transport is allowed through transfer of exactly one particle through the system, and that particle has energy $\Omega_0+\chi$. This is not valid beyond the NESB regime.

Now, let us look at the high temperature regime, $T_1, T_2 \gg \chi, \Omega_0$, where NESB results are not valid. We note that the expression for the currents in Eq.~\ref{currents} has the same form as Eq.~\ref{avg_form}. So, the high temperature trick in Eq.~\ref{avg_form_highT} can be readily applied to obtain
\begin{align}
I^{\tilde{T}\gg \Omega_0,\chi} &\approx \sqrt{\frac{2\chi}{\pi^2\tilde{T}}} \int_0^\infty dx e^{-\frac{ \Omega_0 x +\chi x^2 } {\tilde{T}}} (x+1) \mathcal{I}(\omega_x) ~,\nonumber \\
&\approx\sqrt{\frac{2\chi}{\pi^2\tilde{T}}} \int_0^\infty dx e^{-\frac{ \Omega_0 x +\chi x^2 } {\tilde{T}}} (x+1) A \mathcal{J}(\omega_x) \frac{\Delta T}{\tilde{T}}~, \nonumber \\
J^{\tilde{T}\gg \Omega_0,\chi} &\approx\sqrt{\frac{2\chi}{\pi^2\tilde{T}}} \int_0^\infty dx e^{-\frac{ \Omega_0 x +\chi x^2 } {\tilde{T}}} (x+1)\omega_x A \mathcal{J}(\omega_x) \frac{\Delta T}{\tilde{T}}~,
\end{align}
with $A={\varepsilon^2 \Gamma_1 \Gamma_2}/{(\Gamma_1+\Gamma_2)}$ and $\Delta T = T_1 - T_2$. In the second step above, we have expanded the Bose distributions in $\mathcal{I}(\omega_n)$ (Eq.~\ref{curr_op}) for high temperatures to obtain $\mathcal{I}(\omega_n)\approx
A \mathcal{J}(\omega_n) {\Delta T}/{\tilde{T}}$. Using the general form of spectral function given in Eq.~\ref{J_general}, and after some algebra, we obtain
\begin{align}
\label{currents_highT}
I^{\tilde{T}\gg \Omega_0,\chi} &\approx K\big(s\big) = \frac{A\Delta T}{\sqrt{\pi}\tilde{T}}\int_0^{\infty} dy \Bigg[ y^{-\frac{1}{2}}e^{-y}\left(\sqrt{\frac{\tilde{T}y}{\chi}} + 1\right) \nonumber \\
&\big(\Omega_0+\chi+2\sqrt{\tilde{T}\chi y}\big)^s
\Bigg]  ~,\nonumber \\
J^{\tilde{T}\gg \Omega_0,\chi} &\approx  K(s+1).
\end{align}
For the choice of spectral function in Eq.~\ref{J_general}, we can relate particle and energy currents via the function $K(s)$. We now look at the properties of the function $K(s)$. First, we look at the regime $\tilde{T}\gg \chi \gg \Omega_0$. In this regime, $K(s)$ becomes
\begin{align}
\label{K}
\frac{K\big(s\big)}{\chi^{s-1}\Delta T} &\approx \frac{A}{\sqrt{\pi}}\frac{\chi}{\tilde{T}}\mathcal{F}\big(\frac{\tilde{T}}{\chi},s\big)~, \\
{\rm where}~~\mathcal{F}(z,s)&= \int_0^{\infty} dy \Big[ y^{-\frac{1}{2}}e^{-y}\nonumber \\
&\times \big(\sqrt{zy} + 1\big) \big(1+2\sqrt{zy}\big)^s \Big]~. \nonumber
\end{align}
This then gives
\begin{align}
\frac{K\big(s\big)}{\chi^{s-1}\Delta T} &\approx \frac{2^{s-1}A}{\sqrt{\pi}}\big(\frac{\tilde{T}}{\chi}\big)^{\frac{s-1}{2}}s\mathbf{\Gamma}\left(\frac{s}{2}\right) ,\hspace{10pt}\forall \hspace{5pt} {\tilde{T}\gg\chi\gg\Omega_0}~.
\end{align}
where $\mathbf{\Gamma}(x)$ is the Gamma function.
We immediately make the following observations. Firstly, for $T_1,T_2 \gg \chi \gg \Omega_0$, $K\big(s\big)/(\chi^{s-1}\Delta T)$ varies as a function of ${\tilde{T}}/{\chi}$. Thus, $I/\big(\chi^{s-1}\Delta T\big)$ and $J/\big(\chi^{s}\Delta T\big)$ for various values of $\chi$ should show a data collapse when plotted with ${\tilde{T}}/{\chi}$.

Secondly, we see $K(1)\approx A\Delta T$. It follows that for Ohmic baths ($s=1$ in Eq.~\ref{J_general}) at high temperatures, particle current is independent of the interaction strength $\chi$ and one gets a linear-response-like relation $I\approx A\Delta T$, even for large temperature bias. This is consistent with high temperature result for a harmonic oscillator ($\chi=0$). However, the energy current $J\sim \sqrt{\tilde{T}/\chi} \Delta T $ and shows the effect of interaction. On the other hand for constant bath ($s=0$), the energy current $J$ always satisfies the linear-response-like relation, whereas the particle current is suppressed by a  factor $\sqrt{\chi/\tilde{T}}$.

Thirdly, we note that the quantity $J/(\chi I)$ varies as a function of $\tilde{T}/\chi$ and scales as $\sqrt{\tilde{T}/\chi}$ for any $s$,
\begin{align}
\frac{J}{\chi I}\approx \frac{\mathcal{F}\big(\tilde{T}/\chi, s+1\big)}{\mathcal{F}\big(\tilde{T}/\chi, s\big)}&\approx {2}\sqrt{\frac{\tilde{T}}{\chi}} \hspace{2pt} \frac{(s+1)\mathbf{\Gamma}\left(\frac{s+1}{2}\right)}{s\mathbf{\Gamma}\left(\frac{s}{2}\right)}  \nonumber \\
&\hspace{10pt} \forall \hspace{5pt} T_1,T_2 \gg \chi \gg \Omega_0
\end{align}
In the other regime, $T_1,T_2 \gg \Omega_0 \gg \chi$,  the scaling behaviour is the same, but the data collapse is difficult to see because of large sub-leading terms.

Note that, $I/\Delta T$ and $J/\Delta T$ actually give  the {\emph{beyond linear response}}  analog of particle and energy conductance. For linear response, $\tilde{T}=T$ in the RHS of Eq.~\ref{K}, $T$ being the equilibrium temperature. Then, the above discussion gives the high temperature scaling of conductance.

In Fig.~\ref{fig:currents}, we show the scaling behaviour of $I/\Delta T$ and $J/\Delta T$ as a function of $T_1/\omega_0$. As in Sec.~(\ref{sec:E}), we have chosen $T_1/\omega_0$ as the relevant scaling variable and have kept $r$ fixed at $r<1$ to ensure beyond linear response regime. $r\rightarrow 1$ gives the linear response conductance. The plots (a) and (b) of Fig.~\ref{fig:currents} are then the beyond-linear-response equivalent of temperature scaling of conductance. Note that Eq.~\ref{currents} (and not any simplified expression) was used to calculate currents. The plots show the scaling behaviour discussed above. The plots for $\chi \gg \Omega_0$ show data collapse over the entire range of temperatures. Also, the low temperature behaviour is given by NESB.

In Fig.~\ref{fig:currents}(a), we plot $I/\Delta T$ for fixed $r$ for Ohmic baths ($s=1$ in Eq.~\ref{J_general}).  We see that, for NESB, this quantity behaves non-monotonically with temperature, while for our non-equilibrium SSBH model, this quantity monotonically increases with temperature. In fact, the deviation from NESB result starts precisely at the point where the NESB result reaches a maximum. Similar behaviour is observed for $J/(\Delta T \omega_0)$ in Fig.~\ref{fig:currents}(b).  We conclude that, the non-monotonic behaviour of conductance of NESB, both at and beyond linear response, comes as a result of truncation of the energy spectrum and is not observed when all energy levels are considered. This demonstrates a major difference between the high temperature behaviour of non-equilibrium SSBH model and the NESB model.

Also, it follows from Fig.~\ref{fig:currents}(a), that, for Ohmic baths ($s=1$ in Eq.~\ref{J_general}), at high temperatures the particle current (for fixed $\Delta T$) becomes independent of both interaction strength and the effective temperature. Fig.~\ref{fig:currents}(b) shows a considerable deviation from linear behaviour even for small interaction strengths.

Fig.~\ref{fig:currents}(c) shows that, consistent with our previous discussion, irrespective of the choice of bath spectral function, the quantity $J/(I\omega_0)$, for $\chi \gg \Omega_0$, shows a data collapse and goes as $\sim \sqrt{T_1/\omega_0}$ for $T_1\gg \omega_0$ (with $r$ kept fixed).

Having discussed the scaling behaviour of currents in detail, we now look into another important property of interacting systems, the rectification of current.

\begin{figure}[t]
\includegraphics[scale=0.42]{Fig_3_rect_gamma_2.pdf}
\caption{(color online) The plot shows particle rectification $R_I$ (top) and  energy rectification $R_J$ (bottom) as a function of asymmetry parameter $\gamma$ for various values of interaction strength $\chi$ and Ohmic baths ($s=1$ in Eq.~\ref{J_general}). $R_I$ and $R_J$ are as defined in Eq.~\ref{rectification_def}. The black dotted line corresponds to NESB. The rectfications become maximum when $\gamma\approx 0.6$.  Also, rectification of both particle and energy currents show a non-monotonic change with $\chi$. Rectification for large $\chi$ matches with the NESB result. Energy rectification changes direction with increase in $\chi$. Other parameters are as folows:  $\varepsilon=0.1$, $\omega_c=1000$. All energy variables are measured in units of $\Omega_0$, and time is measured in units of $\Omega_0^{-1}$.
}  \label{fig:three_rect_gamma}
\end{figure}

\subsubsection{Rectification}

Rectification of current is a generic behaviour of non-linear (interacting) systems in non-equilibrium.  As we have seen, in a non-equilibrium set-up, two kinds of currents through the system can be defined, the particle current and the energy current, and  their rectification behaviour can also be quite different. To our knowledge, there has been no previous work where both particle and energy current rectification for a bosonic non-linear system has been investigated. Also, note that rectification can only be observed beyond linear response regime.

Since rectification occurs only for asymmetric system-bath coupling, we use the following definition to describe the degree of asymmetry
\begin{align}
\label{Gamma}
\Gamma_1 = \Lambda(1-\gamma),\hspace{7pt} \Gamma_2 = \Lambda(1+\gamma)~,
\end{align}
where $0 \leq\gamma \leq 1$ is dimensionless. Given a value of asymmetry parameter $\gamma$, we define a measure of rectification as
\begin{align}
\label{rectification_def}
R_I = \frac{I(\Delta T,\gamma)+I(-\Delta T,\gamma)}{I(\Delta T,\gamma=0)}, \hspace{3pt}
R_J = \frac{J(\Delta T,\gamma)+J(-\Delta T,\gamma)}{J(\Delta T,\gamma=0)}~.
\end{align}
$R_I$ and $R_J$ are the particle and energy current rectifications.
This measure of rectification is as used in \cite{SegalPRL2005,SegalJChemPhys2005}. Note that, by this definition, rectification is positive if higher current flows when the cold bath is more strongly coupled to the system. Also, $R_I$ and $R_J$ are zero when $\gamma=0,1$. In our following discussion of rectification, we will primarily confine ourselves to the Ohmic baths ($s=1$ in Eq.~\ref{J_general}).

The variation of $R_I$ and $R_J$ with $\gamma$ is shown in Fig.~\ref{fig:three_rect_gamma}. Irrespective of the strength of interaction strength, we notice that the maximum rectification occurs when $\gamma\approx0.6$. The figure also shows, both particle and energy rectifications behave non-monotonically with $\chi$. For small $\chi$, $R_J$ is negative while $R_I$ is positive, hence, direction of energy rectification is opposite to particle rectification. For large $\chi$, the rectification is same as that obtained from NESB. In the NESB regime, particle and energy rectifications are same, because, particle current is proportional to energy current. All these observations are consistent with our discussion of Fig.~\ref{fig:two_current}.

To concisely investigate the rectification behaviour of the system as a function of the interaction strength and the temperatures, we again resort to the scaling variable $T_1/\omega_0$ with $r$ fixed at $r<1$. The plots of $R_I$ and $R_J$ as a function of $T_1/\omega_0$ for fixed $r$ for Ohmic baths  are shown in Fig.~\ref{fig:rectification}. We readily make the following observations :

Firstly, for $\chi < \Omega_0$, rectification is small. For $\chi \gg \Omega_0$, there is data collapse as expected from the scaling of currents.

Secondly, where NESB matches the non-equilibrium SSBH model, there is small rectification.  This can be understood from expression for currents in NESB regime given in Eq.~\ref{currents_SB}. NESB result holds when $\chi \gg \Omega_0,T_1,T_2$. Therefore, in this regime $\omega_0 \gg T_1,T_2$. So the bose distributions in Eq.~\ref{currents_SB} are exponentially small. Hence, $1+\mathfrak{n}(\omega_0) \approx 1$. With this approximation, the expressions of currents in Eq.~\ref{currents_SB} become antisymmetric under interchange of hot and cold baths. Thus the NESB regime of the SSBH model gives very small rectification. Maximum particle rectification $R_I$ is reached when $\omega_0 < T_1$ which is outside this regime.

Thirdly, after the maximum, $R_I$ gradually approaches zero with increase in $T_1/\omega_0$. This is expected because, as we have seen before, at high temperatures, for Ohmic baths, the particle current eventually has the  form, $I\approx A \Delta T$, which is antisymmetric under interchange of hot and cold baths. On the other hand, the corresponding NESB rectification (the dotted line in Fig.~\ref{fig:rectification}) continues to increase with $T_1/\omega_0$, until it saturates to a high value. Thus, as an effect of having all energy levels, and not truncating at two levels, the particle rectification is suppressed at high temperatures.

\begin{figure}[t]
\includegraphics[scale=0.45]{rectification_scaling.pdf}
\caption{(color online) The plot shows rectification of particle current (top panel) and of energy current (bottom panel) of SSBH model out-of-equilibrium for fixed $r=\frac{T_2}{T_1}=\frac{1}{3}$, for Ohmic baths ($s=1$ in Eq.~\ref{J_general}). $\omega_0=\Omega_0+\chi$. $R_I$ and $R_J$ are as defined in Eq.~\ref{rectification_def}.  The veritcal dashed line indicates the positions of $T_1/\omega_0=1$. The dotted plots correspond to the NESB model. For $\chi \gg \Omega_0$, ($\Omega_0=1$), there is data collapse. Both $R_I$ and $R_J$ has a maximum for $\omega_0<T_1$. $R_J=0$ for $\omega_0\approx T_1$. $R_J$ shows reversal in direction of rectification beyond this point.  Other parameters are as folows:  $\varepsilon=0.1$, $\omega_c=1000$. All energy variables are measured in units of $\Omega_0$ and time is measured in units of $\Omega_0^{-1}$.}  \label{fig:rectification}
\end{figure}

Fourthly, and most interestingly, energy rectification $R_J$ also has a peak  for $\omega_0 < T_1$. However, $R_J=0$ at $\omega_0 \approx T_1$. At this point, the particle rectification is still positive. So the system behaves as a particle rectifier and not as an energy or heat rectifier. Beyond that point, $R_J$ changes sign. Thus the direction of rectification is reversed. With further increase in $T_1/\omega_0$,  energy rectification continues to grow in the reversed direction. Therefore, at high temperatures, the heat rectification occurs in the opposite direction to particle rectification, and continues to grow as temperatures of both hot and cold baths are increased.

Note that all the above observations are for Ohmic baths ($s=1$ in Eq.~\ref{J_general}). Finally, we discuss the case of non-Ohmic baths. For superohmic baths (i.e, for $s>1$), both particle and energy currents show a reversal of direction of rectification. The reversal of energy rectification for superohmic bath occurs at a lower value of $T_1/\omega_0$ than that for the Ohmic bath. For subohmic baths (i.e, for $s<1$), the reversal of energy rectification occurs at higher values of $T_1/\omega_0$ than that for the Ohmic bath. The particle rectification does not show reversal for subohmic baths. For the constant bath, $s=0$, neither particle nor energy rectification shows reversal.  However, in all cases, both particle and energy rectifications vary non-monotonically with the interaction strength.

\subsection{\label{sec:TD}Time dynamics}

Until now, we have discussed the properties of the NESS of the out-of-equilibrium SSBH model. In this section, we look at the transient time dynamics of the various physical quantities we have so far calculated in NESS.

To do this we revert to the equation for time evolution of $\rho_n$, Eq.~\ref{rho_n_equation}. The equation can be re-written and solved in the form
\begin{align}
\label{rho_time}
\frac{\partial\tilde{\rho}(t)}{\partial t}= -\varepsilon^2 M \tilde{\rho}(0) \Rightarrow \tilde{\rho}(t) = e^{-\varepsilon^2 M t}\tilde{\rho}(0)
\end{align}
where $\tilde{\rho}$ is  an infinite dimensional column vector containing diagonal elements of the density matrix and
$M$  is a infinite dimensional non-Hermitian square matrix containing the entries of Eq.~\ref{rho_n_equation}. $M$ has the form
\begin{align}
M=
\begin{bmatrix}
C_0+D_0 & -D_1 &0 &\ldots &\ldots &\ldots\\
-C_0 & C_1+D_1 & -D_2 &0 &\ldots &\ldots\\
0 &-C1 & C_2+D_2 &-D_3 &0 &\ldots\\
\vdots &\ddots &\ddots &\ddots &\ddots &\ddots
\end{bmatrix}
\end{align}
where $C_n$ and $D_n$ are as defined in Eq.~\ref{CD}. Note that the matrix $M$ has the form of a Markov matrix. The sum of each column is zero ($D_0=0$ by definition). This corresponds to the fact that the trace of the density matrix is preserved, i.e, $\sum_n \rho_n = 1$.

To calculate the time dynamics, we choose  an initial state with no particles in the system, i.e, initially, $\rho_0(0)~=~1$, and $\rho_n(0)=0$ $\forall n \neq 0$. The Eq.~\ref{rho_time} is used to numerically obtain the time evolution. Even though the matrices involved are infinite dimensional, for given interaction $\chi$ and temperatures $T_1$ and $T_2$, only a finite number of levels, determined by the ratio of the temperatures and the interaction, effectively contribute. Thus, starting from a finite matrix size, a convergence is reached as the matrix size is increased. Smaller interaction and higher temperatures require larger matrix sizes. A subtle point to note is that, if the matrix $M$ is truncated at any finite size, say $p$, then the constraint that the sum of each column should be zero is not satisfied for the $p$th column, unless $C_p=0$. Consequently, the matrix $M$ can be truncated at size $p$ only if $C_p \ll D_p$.

Since we are using Redfield equation under Born-Markov approximation to obtain transient time dynamics, we need to be careful in choosing the observation times. This is because, as mentioned in Sec.~\ref{sec:C}, Markov approximation is valid only when observation times are much larger than the time for decay of bath correlation functions. We have made some  estimates (similar to that in Ref. \cite{archak}), which indicate that the bath relaxation times are indeed much smaller than the  observational transient times of our interest.

Time dynamics of physical quantities $\langle \hat{N}(t) \rangle $, $\langle \hat{H}_s (t)\rangle $, $I(t)$, $J(t)$ for Ohmic baths ($s=1$ in Eq.~\ref{J_general}) are shown in Fig.~\ref{fig:dyn}, where $I_1(t)$ and $J_1(t)$ are respectively particle and energy currents from the left bath into the system. (Before reaching steady state, the currents from left and right baths are not the same.) Similar time dynamics, but for bosonic system of two sites without interactions, was calculated in Ref.~\cite{archak}.   We observe that, unlike the two site non-interacting case in Ref.~\cite{archak}, here, the currents show no oscillations. More interestingly, we also observe that the time to reach steady state, called $t_{ss}$ hereafter, decreases with increase in the interaction. In the following, we investigate the dependence of $t_{ss}$ on set-up parameters more carefully.

\begin{figure}[t]
\includegraphics[scale=0.42]{dyn.pdf}
\caption{(color online) The figure shows time dynamics of (a) average occupation, $\langle N(t) \rangle$,(b) average energy, $\langle H_s (t) \rangle$, (c) particle current from left bath, $I_1(t)$, and (d) energy current from left bath, $J(t)$, for various values of interaction strength and for a temperature bias $T_1=4$ and $T_2 =2$ for Ohmic baths ($s=1$ in Eq.~\ref{J_general}). All physical quantities demonstrate a non-unitary evolution towards a steady state. The approach to steady state is faster for higher interactions. Since physical quatities plotted here are diagonal in the eigenbasis of the system Hamiltonian, none of them shows oscillations with time. Other parameters are $ \Omega_0~=~1, \varepsilon~=~0.1, \Gamma_1~=~0.4, \Gamma_2~=~1.6, \omega_c~=~1000$. All energy variables are measured in units of $\Omega_0$, and time is measured in units of $\Omega_0^{-1}$.  }  \label{fig:dyn}
\end{figure}

\begin{figure}[t]
\includegraphics[scale=0.45]{dyn2.pdf}
\caption{(color online) The figure shows a log-log plot of time to reach NESS, $t_{ss}$ (as defined in Eq.~\ref{t_ss_def}) as a function of interaction strength $\chi$ for Ohmic baths ($s=1$ in Eq.~\ref{J_general}). The horizontal dash-dotted line corresponds to $\Big[\varepsilon^2\sum_{\ell=1}^2\mathcal{J}_\ell(\Omega_0)\Big]^{-1}$ which is the $t_{ss}$ for the linear ($\chi=0$) system. For large $\chi$, $t_{ss} \approx \Big[\varepsilon^2 \chi \left( \Gamma_1+\Gamma_2\right) \Big]^{-1}$, and this is indicated by the dashed line. For intermediate $\chi$, $t_{ss}$ depends on the temperatures of the hot and cold baths. Other parameters are $ \Omega_0=1, \varepsilon=0.1, \Gamma_1=0.4, \Gamma_2=1.6, \omega_c=1000$. All energy variables are measured in units of $\Omega_0$, and time is measured in units of $\Omega_0^{-1}$.
}  \label{fig:dyn2}
\end{figure}
We note from the solution of $\tilde{\rho}(t)$ in Eq.~\ref{rho_time} that time can be scaled as $\varepsilon^2 t$. It follows that $t_{ss} \propto \varepsilon^{-2}$. Thus, time to reach steady state increases as system-bath coupling becomes weaker.

It is also clear from Eq.~\ref{rho_time} that the steady state is given by the eigenvector of the matrix $M$ corresponding to zero eigenvalue. The fact that a unique steady state is reached in long time then implies that all other eigenvalues of the $M$ have positive real part. Interestingly, we have found in our numerical computation that the eigenvalues of $M$ are all real (and hence no oscillations in time). The smallest non-zero eigenvalue then gives a measure of $t_{ss}$. So, we define
\begin{align}
\label{t_ss_def}
t_{ss}\equiv \frac{1}{\varepsilon^2\lambda_1}~,
\end{align}
$\lambda_1$ being the smallest non-zero eigenvalue of $M$.

Even though $t_{ss}$ cannot be calculated for all set-up parameters analytically, in two limiting cases, analytical results can be obtained. The first case corresponds to the NESB regime $\chi \gg \Omega_0,T_1, T_2$. In this case, $C_1 \ll D_1$, only two levels effectively contribute and matrix $M$ has the form
\begin{align}
M\approx
\begin{bmatrix}
C_0 & -D_1 \\
-C_0 & D_1
\end{bmatrix}
\end{align}
The non-zero eigenvalue of $M$ is
\begin{align}
\lambda_1 = [\varepsilon^2 t_{ss}]^{-1} \approx C_0+D_1 = \sum_{\ell=1}^2 \mathcal{J}_\ell(\omega_0)\left( 2\mathfrak{n}_\ell(\omega_0)+1 \right)
\end{align}
The time to reach steady state therefore also depends on the temperatures of the baths.
However, for large $\omega_0=\Omega_0+\chi$, the Bose distributions are expotentially small.
Thus, for $\chi \gg \Omega_0,T_1,T_2$ and for general bath of the form given in Eq.~\ref{J_general},
\begin{align}
t_{ss} \approx \frac{1}{\varepsilon^2 \chi^{s} \left( \Gamma_1+\Gamma_2\right)}~,
\end{align}
which is independent of the temperatures of the baths. We also see that, for constant baths ($s=0$ in Eq.~\ref{J_general}), $t_{ss}$ is independent of interaction strength. For other baths, $t_{ss}$ decreases with increase of interaction strength as a power law.

The second case where $t_{ss}$ can be analytically, calculated corresponds to the linear system, $\chi=0$. In this case, $C_n$ does not decay with $n$ (Eq.~\ref{CD} for $\chi=0$) and hence the matrix $M$ cannot be truncated at any finite size. So, the above method of finding $t_{ss}$ fails. However, since, in this case, we have a non-interacting system, we can find $t_{ss}$ directly from the evolution equation for $\langle \hat{N}(t) \rangle$. The evolution equation for $\langle \hat{N}(t) \rangle$ can be obtained from Eq.~\ref{N_eqn} by setting $\chi=0$. The resulting equation can be written and solved in the form
\begin{align}
&\frac{d\langle \hat{N}(t) \rangle}{dt} = -\varepsilon^2 \langle \hat{N}(t) \rangle \sum_{\ell=1}^2 \mathcal{J}_\ell(\Omega_0) +\varepsilon^2 \sum_{\ell=1}^2 \mathcal{J}_\ell(\Omega_0)\mathfrak{n}_\ell(\Omega_0) \nonumber \\
&\Rightarrow \langle \hat{N}(t) \rangle = \left( \langle \hat{N}(0) \rangle-N_{ss}\right) e^{-\varepsilon^2 t\sum_{\ell=1}^2\mathcal{J}_\ell(\Omega_0)} + N_{ss}
\end{align}
with $N_{ss}= \Big[\sum_{\ell=1}^2 \mathcal{J}_\ell(\Omega_0)\mathfrak{n}_\ell(\Omega_0)\Big]/\Big[\sum_{\ell=1}^2 \mathcal{J}_\ell(\Omega_0)\Big]$ being the NESS occupation. From above equation, it is clear that for $\chi=0$, the $t_{ss}$ is given by
\begin{align}
t_{ss}=\frac{1}{\varepsilon^2\sum_{\ell=1}^2\mathcal{J}_\ell(\Omega_0)}
\end{align}
This is again independent of the temperatures of the baths.

Except for these two limiting cases, $t_{ss}$ needs to be found numerically using the definition Eq.~\ref{t_ss_def}. Fig.~\ref{fig:dyn2} shows log-log plot of numerically obtained $t_{ss}$ as a function of interaction strength $\chi$ for Ohmic baths for two different choices of temperatures of hot and cold baths. It is seen that except for the limiting cases, $t_{ss}$ depends on the temperatures of the baths. The limiting cases show the behaviour discussed above.

\subsection{\label{sec:G}Conclusion}

We have investigated a system consisting of a single bosonic site with Bose-Hubbard interaction weakly coupled to two bosonic baths at different temperatures. We have used the Redfield QME method to obtain analytical results beyond linear response regime. Below, we summarize our main findings and their potential applications.

We have found an analytical result for the population density (diagonal elements of the density matrix in the eigenbasis of system Hamiltonian) of the system in NESS (Eq.~\ref{rho_n}). This has further allowed us to find various physical observables like average occupation and energy, as well as the particle and energy currents in NESS. We have then analytically found interesting scaling behaviour of the physical observables. Our main finding in this respect is that the high temperature behaviour of the system can be described in terms of an effective temperature (Eq.~\ref{Teff}). Then, it follows that, with the ratio $r$ of the temperatures of cold and hot baths fixed, there occurs a data collapse for various strengths of interaction $\chi$, when physical observables of the system are plotted in terms of the scaling variable $T_1/\omega_0$ (Figs.~\ref{fig:occ_and_energy},\ref{fig:currents},\ref{fig:rectification}). The scaling behaviours hold for a general choice of bath spectral functions of the form given in Eq.~\ref{J_general}. We have also found very interesting rectification behaviour of the system.  The most interesting finding is that, for Ohmic and subohmic baths ($0<s\leq 1$ in Eq.~\ref{J_general}), the energy current shows a reversal in direction of rectification (Fig.~\ref{fig:rectification}). It follows that there is a non-zero strength of interaction strength, ($\chi\approx T_1 - \Omega_0$, for Ohmic baths), where energy or heat rectification is zero. At this point, the system behaves as a particle rectifier but not as a heat rectifier. For superohmic baths ($s > 1$ in Eq.~\ref{J_general}), both particle and energy currents show reversal in direction of rectification. For constant baths ($s=0$ in Eq.~\ref{J_general}), there is no change in direction of rectification for both particle and energy currents. Therefore careful engineering of baths can lead to various interesting rectification behaviour of the system. Such phenomena can be potentially used to create quantum devices, such as, optical diodes.  Reversal of direction of thermal rectification of a quantum system has also been previously theoretically seen for a Heisenberg spin chain out of equilibrium \cite{reversal}. Further, we have computed non-unitary time-dynamics of various physical quantities (Fig.~\ref{fig:dyn}). We found that, except for constant baths, the time to reach steady state is shorter for higher interactions and higher system-bath couplings. For constant baths, with increase in interaction, the time to reach steady state approaches a constant independent of the strength of interaction. For large $\chi$, the time to reach steady state goes as $t_{ss}\sim \big[ \varepsilon^2\chi^{s} \big]^{-1}$ for a general choice of bath spectral functions of the form given in Eq.~\ref{J_general}. All our results also are consistent with the linear, $\chi=0$ (harmonic oscillator) case, as well as the NESB, $\chi \gg \Omega_0, T_1,T_2$, case.

Our results are experimentally relevant in quantum hybrid systems, where a single site Bose-Hubbard model can be realized,  as well as, in molecular junction systems, where our set-up describes a model for anharmonic junctions.  Future work includes going to strong system-bath coupling regime, as well as to generalize to non-equilibrium interacting systems of more than one site, such as, the non-equlibrium Bose Hubbard chain \cite{kordas,biella,hartmann1,hartmann2}  and Jaynes-Cummings Hubbard model \cite{JCH}.

\textit{Acknowledgments:}
AD would like to thank  support   from the Indo-Israel joint research project No. 6-8/2014(IC) and from the  French Ministry of Education through the grant ANR (EDNHS).

\bibliographystyle{unsrt}

redpajama_set_name: RedPajamaArXiv


Are your aerators vanes wearing down and causing the 1423 pumps to fail? If the vanes on your aerator become so worn down that they can no longer pressurize at the necessary levels, your septic system has a serious problem. In order to prevent a complete breakage, these vanes should be replaced immediately. Because buying an entirely new aerator can be rather costly, Gast has come up with an amazing solution to fix your problems, and save you money. By purchasing the Gast K575A 1423 Rebuild Kit, you will be able to look at and fix specific aerator issues instead of investing in a completely new part.
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Without a properly functioning aerator, your septic system can become seriously damaged. Once a septic system stops working correctly, waste will build up and create a harmful environment for you and your family. Ensure your health and safety by purchasing the Gast K575A 1423 Rebuild Kit. With the help of this kit, you can prolong the abilities of your current aerator and cut the cost of splurging on an entirely new piece.
For one low price, this kit includes: 4 vanes, 1 housing gasket, and 2 internal filters. This means that no matter the problem, this rebuild kit has the answer. Remember, an ineffective aerator will be unable to correctly pressurize and break down waste. With this kit, you will never worry about the quality of your air again. Bring your old aerator back to factory specification with the aid of the Gast K575A 1423 Rebuild Kit.
Gast DBMS80-1250 Septic Air Pump Rebuild Kit This Gast DBM80 Repair Kit works for all the following aerators. Gast DBM80-1250 Head assembly which includes frequent wear items diaphragm,check valve, and filter Gast/SSP DBMSs80 repair kit.


Joseph Whitfield Carrillo (born June 11, 1989) is an American Singer-songwriter, and Actor from Santa Clara County, CA, who created the 2018 "Pop, A Cappella" LP album Ultimate Celebrity Blackout. The album's lyrics are against censorship and blackouts of Celebrities hence the albums name. "Ultimate Celebrity Blackout" is about preventing media Blackouts and censorship while promoting free-speech and free-press. Carrillo has also released music videos which he starred in including Don't Black Me Out and You Don't Need School to be a Celebrity, he currently uploads videos on his YouTube channel about various topics including singing and acting.
Ultimate Celebrity Blackout is the first studio album by American singer Joseph Carrillo. It was released on April 17, 2018, through Joseph W Carrillo Music. The Album has 10 original A Cappella songs. Most of the songs have lyrics promoting celebrity, free speech, and free press.
↑ "Election – November city council race gearing up as several seek mayoral seat | Morgan Hill Life".
↑ "Judge keeps ban on ballot box selfies in New York".
↑ "Election: Two incumbents report more campaign contributions - Morgan Hill Times: News".
↑ "Election: PACs, unions load city campaign coffers - Morgan Hill Times: News".
↑ "Election: City candidates report more campaign funds - Morgan Hill Times: News".
↑ "Election: Residents, relatives, developers chime in with campaign contributions - Morgan Hill Times: News".
↑ "Candidates sound off at forums - Morgan Hill Times: News".
↑ "Chamber to host city candidate forum at playhouse - Morgan Hill Times: News".
↑ "Election: Election: Lines drawn on Morgan Hill Measure S - Morgan Hill Times: News".
↑ "Election: Questions and Answers with mayoral candidates - Morgan Hill Times: News".
↑ "Taking the plunge - Morgan Hill Times: News".
↑ "Environmental Extremists Are an Asset to Our Movement".
↑ "Morgan Hill Mayoral Candidate Statement: Joseph Carrillo | Morgan Hill Life".
↑ "Vacant Water District Seat Draws High Level of Interest". 2015-12-04.
↑ "Q&A with Joseph Carrillo, candidate for Morgan Hill City Council - Morgan Hill Times: Home".
↑ "Nov. 4 election: Four vying for two seats on City Council this November | Morgan Hill Life".
↑ "Community story: Candidates announcing runs for various local, county positions | Morgan Hill Life".
↑ "City celebrates grand opening of new council chambers - Morgan Hill Times: City Local Government".
↑ "Freedom Fest events expected to be a big blast | Morgan Hill Life".
↑ "Protesters wave American flags outside school that banned them on Cinco de Mayo".
↑ "Video: Cinco de Mayo Flag Flap Brings Silent Protesters".
↑ "South Bay school readies for Cinco de Mayo protests".
↑ "Cinco de Mayo protests planned for Live Oak HS". 2014-05-05.
↑ "Morgan Hill City Council: Incumbents win two open seats; mayor unopposed – East Bay Times". 2012-08-16.
↑ "Find out who's running for office in Santa Clara County". 2012-08-15.
↑ "Q&A with Council candidate Joseph Carrillo - Morgan Hill Times: Elections".
↑ "Power outage leaves residents hot, in the dark - Morgan Hill Times: Community".
↑ "Morgan Hill City Council results - Gilroy Dispatch".
↑ "Joseph Carrillo responds to questions - Morgan Hill Times: Archives".
This article "Joseph Carrillo" is from Wikipedia. The list of its authors can be seen in its historical and/or its subpage Joseph Carrillo/edithistory. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.


William has two nights at Pitchfork's midwinter festival this upcoming February.
The first on Friday 2/15 with an exclusive performance by The Chicago Philharmonic of The Disintegration Loops at Rubloff Auditorium.
The second on Saturday 2/16 performing his forthcoming record, On Time Out of Time at Fullerton Hall.


text:
Aurland é uma comuna da Noruega, com 1489 km² de área e 1803 habitantes (censo de 2004).

Comunas da Noruega

redpajama_set_name: RedPajamaWikipedia


text:
Q: Spring Boot 2 application.properties in war-deployment on wildfly 14 I have a small spring boot project which should be deployed on a wildfly 14. With spring boot 1 all went well, but after migration to spring boot 2 my settings in application.properties are not used.
I have set spring.main.banner-mode="off" but when the war is deployed it shows the banner in console. When starting the project standalone the banner is hidden.
My application class looks like this:
@SpringBootApplication
public class Application extends SpringBootServletInitializer {

  @Override
  protected SpringApplicationBuilder configure(SpringApplicationBuilder application) {
    return application.sources(Application.class);
  }

  public static void main(String[] args) {
    SpringApplication.run(Application.class, args);
  }

}

And my application.properties is right under src/main/resources.
Does anyone had the same problem or could gibe me an advice how to solve this?

A: Hi is "off" a typing error ?
Please try:
spring.main.banner-mode=off

A: Try
SpringApplication app = new SpringApplication(Application.class);
app.setBannerMode(Mode.OFF);
app.run(args);

Related question : how to disable spring boot logo in stdout?

redpajama_set_name: RedPajamaStackExchange


There really isn't much that can be said for this game that can't be found on any other recap you may come across. Sergei Bobrovsky made the Pens look silly at times, and they couldn't put all three periods together. They came out like a war machine in the 1st, looked flatter than pancakes in the 2nd, and then tried to recover in the 3rd but was just unable to dig themselves out. They put 15 shots on the rookie in the 1st period, and he stopped every single one of them. The mistakes and sloppy play became dominant in the 2nd, and the Pens began to show the rust from the offseason. They came out hard in the 3rd, but couldn't buy a call from the refs and just couldn't get one past Bobrovsky to tie the game up. There was a lot of "excitement," but both teams really lacked the physical play that has come to define these games. To be honest, this game looked more like "croquet on steroids."
I have to say Betts goal late in the 2nd period. It really took the air out of the building, especially since it should have never happened. Mistake by the Pens changing, mistake by Fleury playing the puck, and a mistake by every Penguin on the ice for losing sight of the puck.
Offense- C+ 2 goals against a first-time starter isn't exactly a marvelous accomplishment. They had several opportunities to bury one, but just couldn't get the shot on goal.
Defense- C Nobody really stood out as having a "good game." Michalek was decent, and Martin wasn't bad but everyone else was quite disappointing.
Fleury- B- Ok, so 2 of the 3 goals weren't his fault but he also let like 3 shots behind him that hit the iron. He can do better.
Powerplay- B They moved the puck well, but just had a hard time getting the shot off. Deja vu?
Penalty Kill- B- One major hiccup but they actually looked pretty good after that.
-Despite a few flashes of brilliance, Letang and Goligoski really were plagued by a few glaring mistakes with positioning and turnovers and ended up costing the Pens a few goals. Boy I hope this trend doesn't continue.
-Kennedy looked pretty good, while Talbot and Adams came out flat. This may become significant if Asham returns to the lineup in deciding who to bench.
-This game didn't count for any more points than the next one does. Get back on the horse and go get 'em on Saturday.


text:
VoIP Services
Technology Monthly Newsletter
Tech Tip Tuesday
CTTS C.A.R.E.'s
Schedule Assessment Now
New Report Shows 32 Percent Increase In Cyber Attacks
Posted by cttsonline On July 27, 2018
Positive Technologies has just released a new report that paints a grim picture for IT professionals. If your sense is that the number of cyberattacks are increasing, you're not wrong. In fact, it's probably worse than you realize.
So far this year, we've seen a staggering 32 percent increase in the total number of cyberattacks in the first quarter of 2018, compared to the same period last year. Even worse, malware attacks have increased by a mind-bending 75 percent since the first quarter of 2017.
Individuals were far and away the most frequent target (28 percent), but government agencies came in a reasonably close second (16 percent), with business in the healthcare, finance, and IT sectors rounding out the top five (8 percent, 7 percent, and 5 percent, respectively).
Of the malware deployed against targets this year, spyware was by far the most common, with stealth cryptocurrency mining software not far behind.
Leigh-Anne Galloway, the cyber security resilience lead at Positive Technologies explains in more detail:
"Our research shows that 63 percent of attacks included use of malware. Spyware, in particular, is used most often because it allows obtaining not only personal information and corporate secrets, but credentials for the services and systems needed to attack internal corporate infrastructure."
Long story short, the prevalence of spyware seen in this report is likely to have cast-off effects in the future. Hackers will gain a sufficient level of access and information, courtesy of their successful spyware attacks, to launch even more harmful attacks against both individuals and government/corporate institutions in the future.
Positive Technologies warns that the number of attacks is likely to continue to escalate, and that there's no clear end in sight.
The report is dark news if you're in the digital security business. If your business hasn't been hit yet, it seems that it's just a matter of time.
Apple Adds Physical Security Feature To New MacBook
Watch Your Blog Comments, Or Risk Getting Flagged
Malware In Documents Is Latest Hacker Trend
Ransomware Affected Over 50 Percent Of Surveyed Companies
Reddit Users Advised To Reset Their Passwords
Request A Consultation With Our Award Winning Team
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Protect your business from the latest threats with our weekly Tech Tip Tuesday articles, written by our own Team of experts!
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redpajama_set_name: RedPajamaCommonCrawl


text:
2008 N.Y. Banking Law 441 – Qualifications and disqualification of directors; oath of directors
§ 441. Qualifications and disqualification of directors; oath of directors. 1. At least three-fourths of the directors of the savings and loan bank must reside in the state of New York during their term of office, and all must be citizens of the United States. No person shall be elected a director unless he is the owner in good faith and in his own right on the books of a member savings and loan association of shares having a book value of not less than two hundred dollars and has been nominated by such member savings and loan association for that office; and every person elected to be a director who, after such election, shall hypothecate, pledge or cease to be the owner in his own right of such qualifying shares shall cease to be a director of the savings and loan bank, and his office shall be vacant.
2. Each director, when appointed or elected, shall take an oath that he will, so far as the duty devolves upon him, diligently and honestly administer the affairs of the savings and loan bank, and will not knowingly violate, or willingly permit to be violated, any of the provisions of law applicable to such corporation and that he is the owner in good faith and in his own right on the books of the savings and loan association which nominated him of shares having a book value not less than two hundred dollars, and that the same is not hypothecated, or in any way pledged as security for any loan or debt and, in case of re-election that such shares were not hypothecated or in any way pledged as security for any loan or debt during his previous term. Such oath shall be subscribed by the directors making it, certified by an officer authorized by law to administer oaths, and immediately transmitted to the superintendent.
Gary Kollin, P.A.
Attorney and Lawyer Gary Kollin is an expert and specialist in criminal def…


text:
Q: Laravel: user logs out after session has expired I'm using Laravel 5.5 and the regular provided authentication.
Situation:
The user leaves his computer for 2 hours (the default session timeout) and before leaving the office, he clicks on Logout.
Problem:
The user gets the following error:

The page has expired due to inactivity.
  Please refresh and try again.

Refreshing the page has no effect. The same message is displayed again.
With Laravel 5.4, I had the following error:

TokenMismatchException in VerifyCsrfToken.php (line 68)

The link for logout is in the layout app.blade.php and hasn't been changed:

        Logout

    <form id="logout-form" action="{{ route('logout') }}" method="POST" style="display: none;">
        {{ csrf_field() }}
    </form>

Question
How can I avoid the csrf validation when user clicks on logout?

A: I've just added an exception in App\Http\Middleware\VerifyCsrfToken.php
/**
 * The URIs that should be excluded from CSRF verification.
 *
 * @var array
 */
 protected $except = [
    'logout'
 ];

Ajay Deepak Kumar and Omisakin Oluwatobi are right. This makes the website vulnerable.


Hordes of Argentines gathered in front of the National Congress building in Buenos Aires on Wednesday (3 June) to call for an end to gender-based violence, known as femicide. Similar protests also took place in cities across Argentina, Chile and Uruguay.
Footballer Lionel Messi and Argentina's President Cristina Kirchner led the marchers in Buenos Aires where local media reported 200,000 joined the protest. Women's rights groups, political parties, unions and the Catholic Church have all backed the marches.
Protesters in the city carried banners and wore badges with the campaign slogan Ni Una Menos (Not One Less), and some wore t-shirts with pictures of victims of domestic violence, reports the BBC.
Argentina is among 16 Latin American countries which have passed femicide legislation, introducing harsher legal penalties for those who murder women because of their gender.
The participants at the rally demanded a strengthening of laws and official statistics on violence against women.
According to local women's rights group, Casa del Encuentro, Argentina reported 277 femicides – the killing of a woman by a man because of her gender – in 2014.
Across Argentina, there have been 1,808 femicides since 2008. Back in 2012, the South American nation did pass a law punishing the crime with a life sentence, but the problem shows little sign of abating. Indeed, there have been more than 250 femicides every year since 2010, with a peak of 295 in 2013.
A series of femicide cases have shocked the country.
In April, the estranged husband of a kindergarten teacher stormed into a school where she worked and slit her throat in front of a class in the province of Cordoba.
The boyfriend of a 14-year-old girl is accused of beating her to death when she became pregnant, and in another case a woman was stabbed to death in broad daylight in a café in the capital.


text:
Q: From a url string, how can i get the file path without file name and param var url = 'http ://domain.com/file.php?id=1';
or
var url = 'https ://domain.com/file.php?id=1';
or
var url = 'https ://domain.com';
or
from either one of these urls I want to get only the path, like the one below:
var path = 'https ://domain.com';
and lastly if the url is
var url = 'https ://domain.com/sg';
i should get
var path = 'https ://domain.com/sg';

A: You can try this working example. It should work in your case.
https://jsfiddle.net/ch44Lmuk/10/
function getnewUrl(url) {
    url = url.replace("http://", "");
    var index = url.indexOf('/');
    if (index == -1) return url;
    else {
        var split = url.split('/')
        var domain = split[0];
        if (split[1].length == 0) return domain;
        url = url.slice(index);
        index = url.lastIndexOf('.');

        if (index != -1) {
            index = url.lastIndexOf('/');
            url = url.slice(0, index);
        } else {
            index = url.lastIndexOf('/');
            if (index == url.length - 1) url = url.slice(0, index);
        }
        return domain + url;
    }
}


text:
A combination of CSF tau ratio and midsaggital midbrain-to-pons atrophy for the early diagnosis of progressive supranuclear palsy.
Barbara Borroni, Matteo Malinverno, Fabrizio Gardoni, Mario Grassi, Lucilla Parnetti, Chiara Agosti, Antonella Alberici, Enrico Premi, Ubaldo Bonuccelli, Roberto Gasparotti, Paolo Calabresi, Monica Di Luca, Alessandro Padovani
Journal of Alzheimer's Disease : JAD 2010
Cerebrospinal fluid (CSF) tau ratio decrease (33kDa/55kDa forms) and mid-saggital midbrain-to-pons (MP) atrophy have been suggested as diagnostic markers for progressive supranuclear palsy (PSP). The usefulness of their combined evaluation has never been tested. We evaluated the CSF tau ratio and the MP atrophy as a combined marker for early identification of PSP. A total of 87 subjects, namely 18 PSP, 25 controls (CON), 16 corticobasal syndrome (CBS), and 28 frontotemporal dementia (FTD), were included. Each subject underwent a lumbar puncture and a conventional MRI scan to assess CSF tau 33 kDa/55 kDa ratio and mid-saggital MP measure, respectively. CSF tau ratio and MP ratio were significantly reduced in PSP patients when compared to CON, CBS, and FTD (p< 0.001). Data-based "optimal" combination of CSF tau ratio and MP measure was defined, and the combined marker TrMp= sqrt[3]{CSF Tau ratio}} × {MP ratio} was considered. Considering the combined marker, the difference between the area under the curve (dAUC) of the receiver operating characteristic analysis in PSP versus the various subgroups was higher by about 10% than that obtained by each marker individually. In PSP versus others, a proposed "best" cut-off of TrMP = 0.182 resulted in 94.2% sensitivity and 84.0% specificity. When patients with onset of symptoms ≤ 2 years were included, TrMP resulted significantly decreased in PSP compared to CBS (p< 0.001) and FTD (p< 0.001). The combined marker increases the discriminative power in identifying PSP and suggests that the interplay of different markers should be considered in future trials to enhance diagnostic accuracy from the early stages.


This Keemun Mao Feng is rich and smooth with a light smoke note in the first steeping. Steep lightly to bring out the best of the Keemun notes. This beautiful tea is mildly sweet and smooth with a rich mouth feel.. It is best in the second steeping, and works well with and without milk. Milk removes the wonderful smokey notes of the first steeping.
Keemun is produced in the Qimen County of Huanghsan Shi, in the Anhui province. It has tightly twisted small black leaves with a strong fragrant scent reminiscent of both fruit and roses. Up until the 1870s, the region only produced green teas. At that point, a failed civil servant named Yu Quianchen traveled to the Fujian province to learn their techniques for manufacturing black teas, and then returned to Anhui to apply them. His new tea was excellent, and became extremely popular on the world market. It is used especially in breakfast blends, including our own teahouse blended English Breakfast, Irish Breakfast and Russian Caravan.


Buskaid's upcoming collaboration with Daniel Mille marks a series of 'firsts' for the Soweto String Ensemble. This will be the first time the world famous orchestra plays with an accordionist, the first time they play Astor Piazzolla's music and their first performance at the Orbit.
Over the years Buskaid have collaborated with some very distinguished classical musicians and they are proud to add Daniel Mille to this line-up of international artists. Daniel Mille is one of France's most respected award-winning accordion players. Over the years he performed alongside many a star of the French chanson (Barbara, Claude Nougaro…) as well as playing and recording his own music with the "A Team" of the French jazz scene.
Piazzolla's music is unlike anything else the Buskaid musicians have played before, and its irresistible tango rhythms and sensual improvisatory melodies produced smiles of delight from the young players. From the moment rehearsals started with Daniel Mille, there was the most wonderful air of excitement in the room.
The programme of The Orbit concert will comprise pieces composed by Astor Piazzolla featuring Daniel Mille and the Buskaid string ensemble, pieces by the Buskaid String orchestra and pieces by Daniel Mille with Stéphane Chausse on clarinet.
Daniel Mille and Stéphane Chausse will also perform at the Standard Bank Jazz Festival in Grahamstown on Friday 01 and Saturday 02 July.


Springfield has worked in the modular building industry for over 40 years. We pride ourselves in the unrivalled quality of the portable buildings we offer. Our offsite methods of construction are a combination of modern techniques, tried and tested production methods and a dedication to go above and beyond.
Buying a modular building can give you a permanent solution to any spacing issue and is the perfect choice for a huge variety of businesses. All of the components of our buildings are constructed in our ISO-approved workshop, based in St Helens which has been the source of a vast range of developments all across the UK.
Browse our catalogue of off-the-shelf buildings. These used buildings have been manufactured by Springfield specifically for hire and may be pre-used or new. However, all of the buildings in our catalogue can be reconfigured by our expert craftsman to meet your company's specific needs.
Create a new build to directly meet the specifications of your business. To ensure its feasibility, Springfield is happy to discuss creating a bespoke building with potential buyers. We want to make sure that we fulfil all of your requirements and getting in touch with our dedicated team is the first step in creating your perfect portable building.
Springfield take a unique approach to the buying process for our portable buildings. One of our experienced and professional sales team members will answer a client's initial call and from this point be assigned as their project manager for the entirety of the project. This method provides a client with a single go-to contact who will answer your initial enquiry, establish your needs, produce plans, visit the site and manage the construction and delivery of the development-which includes liaising with any subcontractors.
The first call will be used to qualify the needs of the client and ensure Springfield is the perfect fit for your requirements and that we are able to provide a competitive offer. We are more than happy to handle every aspect of the job offering a bespoke solution if required by the client, which can include handling planning permission, utilities and groundwork contractors.
Why should I buy a portable building?
Springfield is proud of the offsite construction methods we use to construct our portable buildings. The offsite technique involves the construction of your building in our workshop, to then be delivered to site. This process means there is less time spent onsite and more time spent putting the quality touches on your customisable unit. This increase in speed means that your onsite costs will be reduced.
Portable cabins are not the only building type we offer here at Springfield. We also are renowned for our jackleg, prefabricated and multi-storey buildings. We intend to meet our customer's needs and offering a wide range of options is our first step towards such a goal. In addition to the building types, we also offer a fully customisable service for both the exterior finishes of your building and its interior design.
Buying a portable building from Springfield is more than just ordering a building, it's the start of a relationship between supplier and client. We have worked in a vast range of sectors, including and not limited to commercial, sports and leisure, education, and healthcare. Which means we are well-versed in the unique components each sector needs included in their buildings and are able to offer these such as; low level sinks, anti-scald taps for nurseries and fully fitted luxury kitchens in office buildings.
View more of our used portable and modular buildings available to buy or hire.
What Is A Portable Cabin?
What Is A Prefabricated Building?