text:
\section{Introduction}

Critical behavior in quantum optical systems where the particle number is not conserved (see e.g. \cite{Carusotto2009, Kilda19, NoSymmetryBreaking, Baumann2010, Klaers2010}) presents substantial interest as well as conceptual challenges (for a recent discussion see \cite{CiutiSpectralTheorem} and references therein). In order to address the issue, a thermodynamic limit can be formulated via the number of system excitations for which the displayed nonlinearity responsible for criticality can no longer be treated as a negligible perturbation \cite{Carmichael2015}. Light-matter interaction in the form of the Jaynes-Cummings (JC) oscillator, the quintessential model of quantum optics \cite{JCarticle}, is subject to a limit of two kinds on the basis of the above definition, when considered in the presence of drive and dissipation. One of them is a so-called {\it weak-coupling limit}, where quantum fluctuations reduce to an inconsequential noise on top of the semiclassical response. Such a response reminds us of the laser output, in which fluctuations are some orders of magnitude lower than the mean photon occupation of the generated coherence \cite{QO1, EntanglementLaser}. On the other end of the line, this limit pertains as well to the appearance of single-atom resonant absorptive optical bistability \cite{SingleAtom}. The second one, on the other hand, is a {\it strong-coupling limit}, the one born out of a coupling strength, here between a harmonic oscillator and elemental matter in the simplest form of a two-level ``atom'', which appreciably exceeds the dissipation rate.

The breakdown of {\it photon blockade} \textemdash{a} term coined in \cite{Concept97} \textemdash{by} means of a dissipative quantum phase transition, investigated theoretically and experimantally \cite{Carmichael2015, Fink2017, StabilityFSC}, reveals the importance of both these limits and their interplay as we explore the quantum dynamics activated by different sources of dissipation. In particular, the presence of spontaneous emission, even if very small, brings about a competing scaling behavior akin to a weak-coupling limit, gradually eroding the coherence which is manifested by multi-photon resonances and a split Lorentzian, until the whole phase portrait is significantly ``softened'' \cite{Carmichael2015}.

Let us now look at the three interactions comprising our model. Two of them are external and one is internal. The internal interaction refers to the resonant coupling with strength $g$ of the atom with pseudospin raising and lowering operators $\sigma_{\pm}=(1/2)(\sigma_{x}\pm i \sigma_{y})$ and inversion $\sigma_z=2\sigma_{+}\sigma_{-}-1$, and the field mode with annihilation (creation) operator $a$ ($a^{\dagger}$), when these two entities exchange quanta, the familiar Rabi oscillations. One of the external interactions is coherent, namely the coupling to a (semi)classical field with a coherent state amplitude insensitive to quantum fluctuations. This field drives the JC ``molecule'' whose spectrum displays a characteristic $\sqrt{n}$ nonlinearity \cite{Fink2008, Bishop2008}. The other interaction is incoherent, via two channels of dissipation: photon loss through the cavity mirrors with rate $2\kappa$, and spontaneous emission due to coupling of the two-level system to radiation modes other than the cavity mode, with rate $\gamma$. The latter source of incoherence gives rise to a weak-coupling limit, with a system-size parameter scaling as $(\gamma/g)^2$, while the former is linked to a strong-coupling limit subject to a scaling of $(g/\kappa)^2$ \cite{Carmichael2015, SingleAtom}. In the context of these limits, what are we then to expect as the timescale determined by dissipation extends significantly to include the manifestly coherent evolution of such a fundamental quantum nonlinear oscillator?

In this work, we will address this question by focusing on the interplay between the two aforementioned coherent interactions in the strong-coupling limit, as probed via an output channel with a vanishing dissipation rate. Spontaneous emission will be assumed absent unless explicitly stated otherwise. We will first demonstrate the persistence of photon blockade as a manifestly quantum phenomenon, although with a semiclassical flavor, where quantum fluctuations keep on generating a disagreement with the semiclassical picture due to the presence of discrete quantum energy levels against an incoherent evolution. We will then move into these regions of the drive parameter space where quantum-fluctuation switching resolves a neoclassical nonlinearity taking the form of complex-amplitude bimodality; we will find that this resolution is defined by a simple relation between the coupling strength $g$ and a particular configuration $(\Delta\omega, \varepsilon_d)$ of drive detuning and amplitude, respectively. The response here is dominated by bistability with a finite value of the strong-coupling limit scale parameter, the {\it finite-size scaling} of a first-order quantum phase transition in zero dimensions, as explained in \cite{StabilityFSC} : this is a form of instability where quantum fluctuations, owing to the uncertainty principle, cause transitions between two states with a finite lifetime exceeding appreciably the timescale set by dissipation \cite{SingleAtom}. In contrast to the quantum-fluctuation switching that we will observe in the region of quantum blockade, the standard deviation, or uncertainty, over each of the two metastable states becomes visibly smaller than the distance between their means. Finally, we will show that quantum critical behavior foreshadows symmetry breaking when we approach the organizing point of a second-order phase transition on resonance. Here, the importance of quantum fluctuations is gradually diminished as a semiclassical response takes over for $2\varepsilon_d/g \gg 1$.

In particular, after defining the equations governing the quantum and mean-field descriptions in Sec. \ref{sec:MEandNC}, we will consider the relevance of the Kerr oscillator in the development of nonlinearity, and the emergence of resonant {\it quasi} two-level multi-photon complexes in Sec. \ref{sec:photonBl} as we attain the strong-coupling thermodynamic limit. A discussion on the decoherence due to spontaneous emission in Sec. \ref{sec:spontBl} will be followed by an investigation into the regime of smaller detunings and high excitation, where we will present asymptotic expressions for the states of neoclassical bistability in Sec. \ref{sec:CABistanh}, before finally moving to resonance and a dissipative quantum phase transition of second order in Sec. \ref{sec:symmbr}. To that end, we employ semiclassical results, the solution of the master equation (ME) with exact diagonalization in a truncated Hilbert space as well as its unravelling into sample quantum trajectories (see \footnote{We use the {\it Quantum Optics Toolbox} in Matlab for exact diagonalization of the Liouvillian, and the QSD C++ library for the generation of quantum trajectories (quantum state diffusion with adaptive stepsize).} and \cite{qsdreference} for further details).

\section{Master equation and the neoclassical scaling law}
\label{sec:MEandNC}

Within the Born-Markov approximation, the ME determining the evolution of the reduced density matrix $\rho$, namely the dynamics for which the modes of the reservoir the system is coupled to have been lumped into an effective medium with collective attributes, reads \cite{spontdressedstate, QO2}
\begin{equation}\label{MasterEq}
\dot{\rho}=[1/(i\hbar)][H, \rho]+\kappa(2a\rho a^{\dagger}-a^{\dagger}a\rho-\rho a^{\dagger}a),
\end{equation}
with the Hamiltonian in the interaction picture under the rotating wave approximation:
\begin{equation}\label{Hamiltonian}
\begin{aligned}
H=-\hbar \Delta \omega \left(\sigma_{+}\sigma_{-}+ a^{\dagger}a \right)& +i\hbar g \left(a^{\dagger}\sigma_{-}-a\sigma_{+}\right)\\
& + i\hbar\varepsilon_d \left(a^{\dagger}-a\right).
\end{aligned}
\end{equation}
Here, $\Delta\omega=\omega_d-\omega_0$ stands for the detuning, with $\omega_0$ the frequency of the resonant mode coinciding with the two-level spacing of the atom, and $\omega_d$ the frequency of the classical driving field. The first term on the right-hand side of Eq. \eqref{Hamiltonian} gives the occupation of the JC molecule, the second term refers to the coherent {\it JC interaction}, and the third term accounts for the coupling to the external drive with amplitude $\varepsilon_{d}$. The second term on the right-hand side of Eq. \eqref{MasterEq} quantifies the coupling of the resonant cavity mode to a Markovian reservoir taken at zero temperature, populating the output channel. The JC interaction produces {\it dressed states} with frequencies (in the Schr\"{o}dinger picture) $\omega_{0}=0$ and $\omega_{n,\pm}=n\omega_0\pm \sqrt{n}\,g$, $n=1, 2, \ldots$, where the two signs define two ladders of excitation, as pictured in Fig. 1 of \citep{spontdressedstate}. The coherent driving of the JC molecule on resonance ($\Delta\omega=0$) induces a further {\it dressing of the dressed states}, giving rise to the quasi-frequencies (in the interaction picture) $\Omega_{0}=0$ and $\Omega_{m, \pm}=\pm \sqrt{m}\,g[1-(2\varepsilon_d/g)^2]^{3/4}$, $m=1, 2, \ldots$ \cite{quasienergies, QO2, Carmichael2015}. The quasi-energy spectrum at resonance depends on the drive amplitude, and collapses to zero at the critical point $\varepsilon_d=g/2$. It can be extended to account for the JC-Rabi model, with an appropriate renormalization of the drive amplitude \cite{QPTJCRabi, DiracJG2018, RamanDM}.

\begin{figure*}
\begin{center}
\includegraphics[width=\textwidth]{Trajectories_Blockade.png}
\end{center}
\caption{{\it The persistence of photon blockade.} {\bf (a)}-{\bf (c)} Time-dependent photon number averages from single quantum trajectories for a fixed ratio $\varepsilon_{d}/g=0.09$, and time-varying detuning to coupling strength ratio $\Delta\omega(t)/g$ [with $\Delta\omega(t)/g=0.2+t/T$, $0\leq t \leq T$] for $g/\kappa=5, 100, 200$, respectively, superimposed onto the average photon number $\braket{n}_{\rm ss}\equiv\braket{a^{\dagger}a}_{\rm ss}$, as obtained from the steady-state solution of the ME \eqref{MasterEq}. {\bf (d)} $\braket{n}_{\rm ss}$ for $\varepsilon_{d}/g=0.09$ (solid purple line) and $\varepsilon_{d}/g=0.02$ (green dotted line), both with $g/\kappa=5000$. The dashed lines depict the neoclassical response for $\varepsilon_{d}/g=0.09$; past the asterisk on increasing $|\alpha_{\rm ss}|^2$, we give the upper branch of the bistability curve. The dimensionless scan time is $\kappa T=5\times 10^3$ in frame (a) and $\kappa T=5\times 10^2$ in frames (b, c). The insets in frames (a, b) and (c, d) depict contour plots of the steady-state Wigner {\it quasi}-probability distribution $W(x+iy)$ of the intracavity field for a detuning to coupling strength ratio equal to $\Delta\omega/g=1$ and $\Delta\omega/g=0.4510$, respectively. The latter value is very close to the peak of the five-photon resonance, indicated by {\it 5P} in frame (d).}
\label{fig:Blockade}
\end{figure*}
\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth]{5P.png}
\includegraphics[width=0.5\textwidth]{Wigner_6P.png}
\end{center}
\caption{{\it Scanning across an $n$-photon resonance.} \underline{Panel I}: {\bf (a)}-{\bf (c)} {\it Quasi}-probability distributions $W(x+iy)$ of the intracavity field, obtained by the steady-state solution of the ME \eqref{MasterEq}, for a fixed drive to coupling strength ratio $\varepsilon_{d}/g=0.09$ and varying detuning, tracking the five-photon resonance {\it 5P} in Fig. \ref{fig:Blockade}(d). {\bf (d)} Sample quantum trajectory depicting the imaginary part of the intracavity field and the average photon number (displaced by one for clarity), $y(t)={\rm Im}[\braket{a(t)}],\, 1+\braket{n(t)}$, respectively, for the same value of $\Delta \omega/g$ as in frame (c). In (a)-(c), $\Delta\omega/g=0.4500, 0.4520, 0.4526$, and $\braket{n}_{\rm ss}=0.46, 1.49, 1.03$, respectively. \underline{Panel II}: Wigner functions of the intracavity field, obtained by the steady-state solution of the ME \eqref{MasterEq}, for a fixed detuning to coupling strength ratio $\Delta\omega/g=1/\sqrt{6}$ and varying $\varepsilon_d/g$ as $0.048, 0.108, 0.168, 0.196$ in frames {\bf (a)}-{\bf (d)}, respectively. In both panels, $g/\kappa=5000$.}
\label{fig:5P6PWigner}
\end{figure}

Considering average values of the corresponding to the ME \eqref{MasterEq} Heisenberg equations, and factorizing the expectations of the operator products yields the {\it neoclassical} equations of motion. Solving then for the steady-state intracavity field, $\alpha_{\rm ss}$, with negative two-level inversion, we obtain an equation for its modulus, featuring the {\it strong-coupling scaling parameter} $n_{\rm sc}=g^2/(4\kappa^2)$ \cite{Carmichael2015},
\begin{equation}\label{neocl}
|\alpha_{\rm ss}|^2=(\varepsilon_{d}/\kappa)^2\,f(\Delta\omega/\kappa,g/\kappa;|\alpha_{\rm ss}|^2/n_{sc}),
\end{equation}
with
\begin{equation*}
\begin{aligned}
f(\Delta\omega/\kappa,g/\kappa;&|\alpha_{\rm ss}|^2/n_{\rm sc})=\{1 +[\Delta\omega/\kappa -{\rm sgn}(\Delta\omega)\\
&\times (\Delta\omega^2\kappa^2/g^4 +|\alpha_{\rm ss}|^2/n_{\rm sc})^{-1/2}]^2\}^{-1},
\end{aligned}
\end{equation*}
a nonlinear function of the scaled neoclassical cavity-field amplitude. Coming from that origin, in what follows we will investigate the correspondence between the neoclassical scaling law of Eq. \eqref{neocl} and the demonstration of critical behavior in the quantum picture as we attain the thermodynamic limit in various regions of the drive parameter space $(\Delta\omega/g, \varepsilon_{d}/g)$.

\section{From semiclassical perturbation theory to the discrete JC spectrum}
\label{sec:photonBl}

\subsection{Mapping to the Kerr oscillator}
\label{subsec:mappringKerr}

We will now track the emergence of low-amplitude nonlinearity due to the atom-field coupling, taking the form of a deviation from a Lorentzian profile. For a very small excitation, with $2g^2|\alpha_{\rm ss}|^2 \ll \Delta\omega^2$, the nonlinearity can at first be neglected, yielding a response which peaks for the two Rabi resonances at $\Delta\omega/g=\pm 1$. Further along, the first occurrence of nonlinearity is of Kerr-type, following the expansion of the square root in Eq. \eqref{neocl} to first order in $|\alpha_{\rm ss}|^2/n_{\rm K}$, with $n_{\rm K}\equiv\Delta\omega^2/(2g^2)$. A new scaling emerges for the first appearance of bistability, independent of dissipation and akin to a weak-coupling limit. In the development of the vacuum Rabi resonance with increasing $n_{\rm sc}$, for $\Delta\omega=g$ and $|\alpha_{\rm ss}|^2 \ll 1/2$, the neoclassical equation precludes any low-amplitude bistability, as the intracavity excitation reads
\begin{equation}\label{vacRessemi}
|\alpha_{\rm ss}|^2 \approx \varepsilon_{d}^2 (\kappa^2+4g^2|\alpha_{\rm ss}|^4)^{-1},
\end{equation}
mapped to an effective Kerr oscillator with zero detuning. Taking $\kappa^2 \ll 4|\alpha_{\rm ss}|^4 g^2$ in a self-consistent fashion, we obtain $|\alpha_{\rm ss}|^2 \approx [\varepsilon_d/(2g)]^{2/3}$, valid for $\varepsilon_d/(2g) \ll 1$. When $\Delta\omega<g$ (with $\Delta\omega>0$), however, the detuning of the equivalent Kerr oscillator, $\overline{\Delta \omega}=\Delta\omega-g^2/\Delta\omega$, will eventually satisfy the condition of bistability, $(\overline{\Delta \omega})^2>3 \kappa^2$ [with $\overline{\Delta \omega}\,(g^4/\Delta\omega^3)<0$], in the limit where the dissipation rate tends to zero (see Eq. (32) of \citep{Carmichael2015} and Sec. 2 of \cite{Drummond_1980}). We note that since $n_{\rm K} \sim 1/2$ in the region of the vacuum Rabi resonance, we expect that quantum fluctuations will produce visible deviations from the mean-field nonlinearity, already for a very weak intracavity excitation.

When the condition $2g^2|\alpha_{\rm ss}|^2 \ll \Delta\omega^2$ is not upheld, for larger excitations or for smaller detunings, the perturbative treatment of the neoclassical nonlinearity discussed above is no longer valid. Moreover, in the limit $\Delta\omega/\kappa\to 0$, a divergence is encountered for $|\alpha_{\rm ss}|\to 0$, marking the inadequacy of the semiclassical theory and the need to explicitly bring quantum levels into consideration \cite{Carmichael2015}.

In Fig. \ref{fig:Blockade}, individual quantum trajectories map out the emergence of the characteristic $\sqrt{n}$ nonlinearity through a spectrum of non-equidistant resonances for a time-varying positive detuning, with growing $n_{\rm sc}$. For a dissipation rate $\kappa$ which is of the order of the coupling strength $g$, only a broad (vacuum Rabi) resonance is present, centered at $\Delta\omega=g$ [frame (a)]. When $n_{\rm sc}$ grows, however, the single-photon transition saturates and multi-photon resonances \cite{multiphotonres} are revealed, with varying weight, some {\it pre} and others {\it post} saturation, in the driving frequency response [frames (b-d)]. The saturation of the vacuum Rabi resonance in the quantum dynamics, $\braket{a^{\dagger}a}_{\rm ss, \,max} \approx 1/4$ (see Fig. 3 of \cite{effectivetwolevel}), with a more obvious separate peak for a lower value of $\varepsilon_d/g=0.02$ and $\varepsilon_d/\kappa=100$, as shown in Fig. \ref{fig:Blockade}(d), takes us away from the validity region of the semiclassical mapping to the Kerr oscillator and the prediction $|\alpha_{\rm ss}|^2 \approx [\varepsilon_d/(2g)]^{2/3}$ of Eq. \eqref{vacRessemi} \textemdash{the} ratio $\varepsilon_d/\kappa$ must now be taken into account in the nonlinearity of saturation.

The mean-field perturbative prediction is invalidated for a further decreasing dissipation rate and the ensuing growth of the intracavity excitation; the steady-state photon number average in the quantum picture proceeds first as a monotonically increasing and then a constant function of the ratio $\varepsilon_{d}/\kappa$ in the thermodynamic limit. The average photon number $\braket{a^{\dagger}a}_{\rm ss}$ is fixed at $1/4$ as long as $\varepsilon_{d}/g$ remains sufficiently small. Moreover, the mapping in question places the onset of bistability at
\begin{equation}\label{DuffingDw}
\Delta\omega/g \approx \{[1+(1-\sqrt{3}\kappa/g)^2]/2\}^{1/2}
\end{equation}
in the strong-coupling limit, approaching the vacuum Rabi resonance with very low excitation; this prediction of Eq. \eqref{DuffingDw} is in contrast to what we observe in Fig. \ref{fig:Blockade}(d). Nevertheless, the heralded bistability, pertubatively of Kerr-type, will remain of relevance in our discussion below, as anticipated by instances of quantum bistable switching around the vacuum Rabi resonance in the trajectories of Figs. \ref{fig:Blockade}(b, c), which are absent from the resonance of Fig. \ref{fig:Blockade}(a).

\subsection{Emergence and persistence of photon blockade}
\label{subsec:emergpb}

In order to account for the thermodynamic limit with respect to photon blockade for a large $n$, we read in \cite{Carmichael2015} that an $n$-photon resonance occurs for $\Delta\omega=\pm g/\sqrt{n}$, while the ($n+1$)-photon resonance is ``blocked'' by an effective detuning $(\omega_{n+1,\pm}-\omega_{n,\pm})-\omega_{d} \approx \mp \kappa\sqrt{n_{\rm sc}/n}$. To arrive at this expression, we have used a Taylor expansion of the square root in $\pm g(\sqrt{n+1}-\sqrt{n}) \approx \pm g/(2\sqrt{n})$ \citep{spontdressedstate}, and substituted for $g$ from the definition of $n_{\rm sc}$. At the same time, Eq. \eqref{neocl} provides an indication on the extent of semiclassical nonlinearity through an offsetting detuning reading
\begin{equation}\label{detepskappa}
(\Delta\omega/g)^2=2(\varepsilon_d/\kappa)^2(\sqrt{1+(1/4)(\kappa/\varepsilon_d)^4}-1),
\end{equation}
to produce $|\alpha_{\rm ss}|^2=(\varepsilon_d/\kappa)^2$. Both regimes, that of photon blockade and that of neoclassical bistability, are extended and contrasted for $\kappa/\varepsilon_d\to 0, \kappa/g\to 0$, while for large values of the ratio $\varepsilon_d/\kappa$ we move towards the boundary $\Delta\omega/g=\pm\kappa/(2\varepsilon_d)$ where individual transitions are not resolvable, as we will discuss in Sec. \ref{sec:CABistanh}. Conversely, for $\varepsilon_d/\kappa \ll 1$ from Eq. \eqref{detepskappa} we obtain $\Delta\omega/g=\pm 1$, relevant to the build-up of the vacuum Rabi resonance we met in Sec. \ref{subsec:mappringKerr}. An appropriate choice for the scaled drive, reflecting the impact of nonlinearity in the strong-coupling limit, would then be $\sqrt{(\varepsilon_d/\kappa)^2/n_{\rm sc}}=2 \varepsilon_d/g$.

As $n_{\rm sc}$ is varied from $\sim 1$ to $\sim 10^6$ in frames (a-d) of Fig. \ref{fig:Blockade}, we find that the effective detuning does not compromise the resonances formed for successive values of $n$, while the regime of multi-photon blockade expands to include about nine visible resonance peaks whose location is determined predominantly by the discrete level structure. The Wigner functions drawn in the insets of frames (a, b) correspond to the peak of the vacuum Rabi resonance, displaying a mild nonlinearity with respect to the vacuum state. On the other hand, the Wigner functions in frames (c, d) evidence the onset of pronounced quantum fluctuations intertwined with the displayed low-photon nonlinearity \cite{QO2} in the region of the five-photon resonance with a peak expected at $\Delta\omega/g=1/\sqrt{5}\approx 0.4472$.

It then becomes clear that fluctuations do not vanish as $n_{\rm sc}$ is taken to infinity and $|\alpha_{\rm ss}|^2/n_{\rm sc}$ itself vanishes. Rather, they reveal conspicuously the discrete energy spectrum of the JC interaction. For growing $n$, the successive $n$-photon resonances become increasingly sharper, and their width approaches the order of magnitude of $\kappa$, where $\kappa/g\to 0$ (for $n_{\rm sc}\to \infty$). In this regime, low-amplitude bistability with large fluctuations accompanies the buildup of a multi-photon resonance, as we can see in the Wigner functions of Fig. \ref{fig:5P6PWigner}I(a-c) drawn for varying detuning when scanning across the sharp {\it post} saturation five-photon resonance peak {\it 5P}. The quantum trajectory for the intracavity field in frame (d) of Panel I shows quantum-fluctuation switching between a vacuum state \footnote{also frequently called {\it dim state}, as its photon average can be very small in comparison to that of the excited state yet nonzero.} and an excited state with mean amplitude of modulus squared $|\alpha|^2 \approx 1.5$, abiding with the prediction of Eq. \eqref{neocl}. The emergence of neoclassical bistability with growing $\varepsilon_d/g$ is also evident for a detuning where one would expect the peak of the six-photon resonance, $\Delta\omega/g=1/\sqrt{6} \approx 0.4082$, as depicted in Fig. \ref{fig:5P6PWigner}II. In the Wigner function of frame (d), for example, instead of the six-photon state, one finds an attractor with a mean amplitude of modulus squared $|\alpha|^2 \approx 2.7$, a value close to the upper branch of the mean-field bistability curve.

Bistability features in photon blockade in spite of the obvious disagreement between quantum dynamics and the mean-field response. In Panel I of Fig. \ref{fig:channel}, we keep $g$ and $\varepsilon_{d}$ constant, with $\varepsilon_{d}/g=0.18$, and increase $\kappa$ [in the spirit of Fig. \ref{fig:Blockade}] when plotting the three curves (i)-(iii) depicting the steady-state average photon number. Comparison between curve (i) of Fig. \ref{fig:channel}I and frame (d) of Fig. \ref{fig:Blockade} demonstrates that resonances become sharper and occur closer to the value $\Delta\omega/g=\pm 1/\sqrt{n}$ with decreasing $\varepsilon_{d}/g$, provided that $n_{\rm sc}$ is large enough. Curves (i)-(iii) share the same value of $\varepsilon_{d}/g$, double of that used for Fig. \ref{fig:Blockade}, with decreasing values of $n_{\rm sc}$. Their difference is appreciable only in the multi-photon resonances with $n\geq 7$, where a larger $n_{\rm sc}$ signifies sharper resonances. This is in contrast to Fig. \ref{fig:Blockade}, where such a disparity occurs for lower values of $n$ for the same values of $n_{\rm sc}$.
\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth]{lowerg.png}
\includegraphics[width=0.5\textwidth]{Qfuncs.png}
\includegraphics[width=0.5\textwidth]{GammaBist.png}
\end{center}
\caption{{\it Complex-amplitude bimodality with higher photon numbers.} \underline{Panel I}: Steady-state average intracavity photon number obtained by the solution of the ME \eqref{MasterEq} for a fixed ratio $\varepsilon_{d}/g=0.18$, $\varepsilon_{d}/\kappa=900, 90, 45$, (and $g/\kappa=5000, 500, 250$) for the curves (i) [in solid purple], (ii) [in blue dots] and (iii) [in orange asterisks], respectively. The insets A, B depict the steady-state Wigner function $W(x+iy)$ for $\Delta\omega/g=0.392, 0.408$, respectively, along curve (iii). The corresponding seven-photon resonance is indicated by {\it 7P}. \underline{Panel II}: {\it Quasi}-probability distributions $Q(x+iy)$ of the intracavity field, obtained by the steady-state solution of the ME, for a fixed drive to coupling strength ratio $\varepsilon_d/g=0.245$, a fixed detuning to coupling strength ratio $\Delta\omega/g=0.1$, and a varying ratio $g/\kappa=10, 20, 10^2, 10^4$ in frames {\bf (a)}-{\bf (d)}, respectively. \underline{Panel III}: Steady-state {\it quasi}-probability distributions $Q(x+iy)$ of the intracavity field for $\varepsilon_d/g=0.245$, $\Delta\omega/g=0.1$ and $g/\kappa=10^2, 10^4$ in frames {\bf (a)}, {\bf (b)}, respectively, in the presence of spontaneous emission with rate $\gamma/\kappa=1$.}
\label{fig:channel}
\end{figure}

We then proceed to plot the Wigner functions of the intracavity field in the two insets of Panel I in Fig. \ref{fig:channel}, for a dip and the peak shaping the seven-photon resonance $7P$ along curve (iii). Quantum fluctuations organizing neoclassical bistability cannot secure a good representation of the vacuum state, as was the case in Panel I of Fig. \ref{fig:5P6PWigner}, resulting in softened features \footnote{Semiclassical bistability in Panel I of Fig. \ref{fig:channel} sets in for $\Delta\omega/g \approx 0.59$, past the first few resonances \textemdash{compare} with Fig. \ref{fig:Blockade}(d). We also note that for a given (nonzero) value of $\Delta\omega/g$ and a large enough $n_{\rm sc}$, the neoclassical bistability curve against $\varepsilon_d/g$ is not affected by a further decrease of the dissipation rate, as Eq. \eqref{modulusgeps} suggests for higher excitation.}. Returning to Fig. \ref{fig:Blockade}(d), we find that the enhancement of bimodality with respect to the varying representation of the vacuum and excited states across an $n$-photon resonance close to saturation can be seen as a competition between the JC interaction and coupling to the external field under vanishing dissipation. Therefore, in the thermodynamic limit and provided that $\varepsilon_d/\kappa$ is appreciably larger than unity, promoting $g$ exposes the quantum energy scale against the breakdown of photon blockade, while a growing $\varepsilon_{d}$ saturates fast the $n$-photon transition before it merges with the $(n+1)$-peak and eventually disperses into a continuum of unresolved resonances. This gradually brings us to the regime of large intracavity photon numbers approaching the order of a diverging $n_{\rm sc}$ as optical bimodality \textemdash{the} manifestation of a semiclassical nonlinearity that cannot be regarded as a negligible perturbation \textemdash{sets} in; this region is discussed in Sec. \ref{sec:CABistanh}. In that sense, the multi-photon transitions responsible for the nonlinear response acquire now a more continuous and classical dimension, in contrast to the regime of photon blockade where raising $n_{\rm sc}$ reveals more resonances as unabated quantum fluctuations.

\section{Spontaneous emission in the strong-coupling limit}
\label{sec:spontBl}

Before joining the two regions, we will make a brief detour to comment on the r\^{o}le played by spontaneous emission. Its inclusion adds the term $(\gamma/2)(2\sigma_{-}\rho \sigma_{+}-\sigma_{+}\sigma_{-}\rho-\rho\sigma_{+}\sigma_{-})$ on the right-hand side of ME \eqref{MasterEq} and breaks the conservation law for the pseudospin, introducing a new semiclassical scaling relation akin to a weak-coupling limit. The complex intracavity mean-field amplitude is written as \cite{Carmichael2015}
\begin{equation}\label{GammaOB}
\alpha_{\rm ss}=(\varepsilon_d/\kappa)\, h(\Delta\omega/\kappa, \tilde{\gamma}/\kappa, g^2/|\tilde{\gamma}|^2, |\alpha_{\rm ss}|^2/\tilde{n}_{\rm wc}),
\end{equation}
where
\begin{equation*}
\begin{aligned}
&h(\Delta\omega/\kappa, \tilde{\gamma}/\kappa, g^2/|\tilde{\gamma}|^2, |\alpha_{\rm ss}|^2/\tilde{n}_{\rm wc})=\{1-i\Delta\omega/\kappa \\
&+ [g^2\tilde{\gamma}/(\kappa |\tilde{\gamma}|^2)](1+|\alpha_{\rm ss}|^2/\tilde{n}_{\rm wc})^{-1}\}^{-1},
\end{aligned}
\end{equation*}
with $\tilde{\gamma}\equiv \gamma/2+i\Delta\omega$ and $\tilde{n}_{\rm wc}\equiv|\tilde{\gamma}|^2/(2g^2)$. This produces the Maxwell-Bloch scaling law corresponding to absorptive optical bistability on resonance ($\Delta\omega/g=0$), as mentioned in the Introduction, together with the weak-coupling scaling parameter $n_{\rm wc}=\gamma^2/(8g^2)$ that we will make use of to quantify decoherence. For $\gamma=0$, it yields the scaling parameter $n_{\rm K}$ we have already met in Sec. \ref{subsec:mappringKerr} when making the semiclassical perturbative mapping to the Kerr oscillator, valid only for very low excitation. A detailed comparison between the quantum and semiclassical pictures in the strong-coupling limit with spontaneous emission present lies beyond the scope of our work. Nonetheless, we may note straight away that coupling of the two-level atom to the radiation modes of the reservoir has an ostensible impact on quantum coherence and the low-amplitude bistability associated with photon blockade.
\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth]{WignerGamma.png}
\includegraphics[width=0.475\textwidth]{nGamma.png}
\end{center}
\caption{{\it Spontaneous emission and photon blockade.} Wigner distributions $W(x+iy)$ of the intracavity field, obtained by the steady-state solution of the ME \eqref{MasterEq}, for a fixed detuning and drive to coupling strength ratio $\Delta\omega/g=0.4132$ and $\varepsilon_{d}/g=0.1$, respectively, in the presence of spontaneous emission with $\gamma/\kappa=0.2, 2, 20, 200$ in frames {\bf (a)}-{\bf (d)}, respectively. {\bf (e)} Steady-state intracavity photon number for variable detuning, obtained from the solution of Eq. \eqref{MasterEq} in the presence of spontaneous emission, for $\varepsilon_{d}/g=0.1$ and $\gamma/\kappa= 2, 20, 200$ in the curves (i) [solid purple], (ii) [orange dots], (iii) [green crosses], respectively. The five (six)-photon resonance peak is denoted by {\it 5P} ({\it 6P}). The inset depicts the Wigner distribution of the intracavity field for the same driving point as in frames (a-d) but with $\gamma/\kappa=3$. In all frames, $g/\kappa=5000$.}
\label{fig:GammaWigner}
\end{figure}

In frames (a-d) of Fig. \ref{fig:GammaWigner}, we follow the decay of the six-photon resonance peak {\it 6P} as $n_{\rm wc}$ is varied from $\sim 10^{-10}$ to $\sim 10^{-4}$. With increasing $\gamma/\kappa$, the $n$-photon resonances are compromised while their relative height changes to favor lower values of $n$ [frame (e)] as a result of the inability to sustain coherence over a many-photon process. More than the expected broadening of the quasi-probability distribution with pronounced quantum fluctuations, we observe a transition from the bright metastable state to the vacuum state as the photon occupation is lowered. This evidences once more the underlying connection of bistability and photon blockade; the peak height, which subsequently falls, is again in correspondence to the excited neoclassical state we met in Fig. \ref{fig:5P6PWigner} II(d). It is, however, quantum fluctuations that efface bistability in a manner distinct to merely altering the switching rate between two metastable states, as dictated by the semiclassical nonlinear dynamics. We will take this discussion further in the next section.

\section{From complex-amplitude bimodality to the anharmonic-ladder oscillator}
\label{sec:CABistanh}

We will now turn to the higher cavity-field amplitudes, following the breakdown of photon blockade by means of a first-order quantum phase transition \cite{Carmichael2015}. Here quantum fluctuations induce switching between states differing by a substantial amount of photon occupation. We can get a good impression of the approach to the strong-coupling limit from the {\it quasi}-probability distributions $Q(x+iy)$ of the intracavity field, as depicted in Panel II of Fig. \ref{fig:channel}. In this region of the drive parameter space, quantum fluctuations essentially restore the neoclassical nonlinearity, with a development of complex-amplitude bimodality for increasing $n_{\rm sc}$. In frames (a-d), we witness the birth of a highly-excited metastable state out of the vacuum state in the $Q$ function for the intracavity field, which eventually assumes most of the excitation probability following a spiral rotation of the mean complex amplitude in the first quadrant of the phase space.

When $4 g^2|\alpha_{\rm ss}|^2 \gg \Delta\omega^2$, i.e., for the reverse asymptotic behavior to the one considered in Sec. \ref{subsec:mappringKerr}, we find two roots of the neoclassical equation,
\begin{equation}\label{modulusgeps}
|\alpha_{\rm ss, \pm}|^2\approx [(g\pm 2\varepsilon_d)/(2\Delta\omega)]^2,
\end{equation}
as $\kappa \to 0$, while there is no mean-field bistability for $\varepsilon_d > g/2$. Although $|\alpha_{\rm ss, +}|$ in Eq. \eqref{modulusgeps} sets an upper boundary for the cavity excitation (see e.g. the successive excited metastable states in Fig. 3(a) of \cite{StabilityFSC} for increasing $g/\kappa$), it is still up to the quantum fluctuations to determine the balance between the two metastable states \textemdash{a} situation pictured in frame (d) of Panel II \textemdash{and} consequently the value of $\braket{a^{\dagger}a}_{\rm ss}$. What is perhaps more illuminating regarding the coherent superposition of the drive field with amplitude $\varepsilon_d$ and the field re-radiated by the two-level atom, with amplitude $g\braket{\sigma_{-}}_{\rm ss, +} \to g/2$, is the other root of the neoclassical equation in the limit under consideration, $|\alpha_{\rm ss, -}|^2\approx [(g-2\varepsilon_d)/(2\Delta\omega)]^2$, corresponding to the unstable branch of the bistability curve \textemdash{a} destructive interference between the two fields. This expression produces the form $0/0$ on resonance at the critical point. At the ratio $g/\kappa \approx 43$, a value between $20$ and $100$ [in frames (b) and (c) respectively], we cross the boundary of the first-order dissipative quantum phase transition; for the corresponding value of $n_{\rm sc}=n_{\rm sc, b}$ as determined by quantum fluctuations, the peak heights of the two states are equal in the $Q$ function. On approaching resonance, $\Delta\omega/g \to 0$, the two states form an angle $\theta_{\pm} \approx \pm \sqrt{n_{\rm sc}/|\alpha_{\rm ss}|^2}$ with respect to the $x$-axis for $2 \varepsilon_d/g \gg 1$, a limit where the JC interaction can be treated as a perturbation [see Eq. \eqref{thetass} in Sec. \ref{sec:symmbr}].
\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth]{FigQubit.png}
\end{center}
\caption{{\it On the boundary of a first-order dissipative quantum phase transition.} Sample quantum trajectory depicting $Z(t) \equiv \braket{\sigma_z(t)}$ in frame {\bf (a)} and the projection of atomic coherence (as a distribution) on the equatorial plane of the Bloch sphere [with $(X,Y)(t) \equiv \braket{\sigma_{(x,y)}(t)}$] in frame {\bf (b)}. The inset depicts a frequency histogram (with a large bin number) for the inversion $Z(t)$, as extracted from the sample trajectory. Here, $g/\kappa=43.48$, with the remaining parameters being identical to those used for Panel II of Fig. \ref{fig:channel}.}
\label{fig:qubitqpt}
\end{figure}
In Fig. \ref{fig:qubitqpt}, we depict quantum-fluctuation switching between equiprobable metastable states \footnote{The qualification of equal probability is used in a loose sense here, since the condition of equal areas under the two peaks in the {\it quasi}-probability distribution does not also mean equal peak heights in the presence of quantum fluctuations for a finite-size scaling \textemdash{see} Sec. VIII of [Bonifacio R. {\it et al.}, Phys. Rev. A {\bf 18}, 2266 (1978)] considering as well the weak-coupling limit of many-atom absorptive resonant bistability (with fluctuations scaled by the number of atoms in the cavity).} in a sample realization for the atomic inversion next to the coherence $2\braket{\sigma_{-}(t)}$ for $n_{\rm sc, b}$. The projection of the distribution corresponding to the excited state is bounded by the circumference of the circle $|2\braket{\sigma_{-}(t)}|=1$ on the equatorial plane, in the limit $4 g^2|\alpha_{\rm ss}|^2 \gg \Delta\omega^2$, while the projection corresponding to the vacuum state lies inside the circle, as pictured in frame (b) [see also Eq. (17) of \cite{Carmichael2015}]. On resonance, the two equiprobable distributions are instead symmetrically positioned on the equatorial plane.

The multi-photon resonances we met in Sec. \ref{subsec:emergpb} are smeared out as the intracavity amplitude grows with vanishing $\kappa$, and eventually merge into the ``tail-envelope'' of a split Lorentzian peaking at $\Delta \omega/g=\pm \kappa/(2\varepsilon_{d})$ in the direction of resonance, with $|\alpha_{\rm ss}|^2=(\varepsilon_{d}/\kappa)^2$. The effective detuning contains $\Delta\omega$ as a free variable of the split Lorentzian function, with $(\omega_{n+1,\pm}-\omega_{n,\pm})-\omega_{d} \approx -\Delta\omega\pm g/(2\sqrt{n})$, and is not considered in reference to the discrete steps along the two ladders for which the scaling relationship of photon blockade was formulated. In that nonlinear response, the disparity between neoclassical theory and quantum dynamics recedes with increasing $\varepsilon_d/\kappa$ (see, e.g., Fig. 4 of \citep{Carmichael2015}).

Prompted by the emerging agreement between the two pictures and leaving bimodality aside for the moment, we set $\Delta\omega=g/\sqrt{\braket{n}}$ in Eq. \eqref{neocl} [hence selecting the upper excitation ladder without loss of generality, while retaining the semisclassical nonlinearity in the response (which is canceled at $\Delta\omega=g/\sqrt{2\braket{n}}$)], with $\braket{n} \equiv \braket{a^{\dagger}a}_{\rm ss}$ treated as a continuous variable. This yields
\begin{equation}\label{UpperOsc}
\braket{n} + (g/\kappa)^2[1-\braket{n}(4\braket{n}^2+1)^{-1/2}]^2=(\varepsilon_{d}/\kappa)^2,
\end{equation}
giving $\braket{n} \approx (\varepsilon_{d}/\kappa)^2-n_{\rm sc}=n_{\rm sc}[(2\varepsilon_{d}/g)^2-1]$ above threshold on approaching the thermodynamic limit with high excitation \textemdash{a} benchmark of the empty-cavity excitation with respect to the scaling photon number. At the mean-field level, this result brings us to the model of the quantum $\sqrt{n}$ anharmonic oscillator on resonance, formulated via either of the two excitation ladders of the JC spectrum (see also Sec. 16.3.5 of \citep{QO2}). As we will see in Sec. \ref{sec:symmbr}, however, the r\^{o}le of quantum fluctuations is special about the critical point of the JC second-order quantum phase transition. The presence of semiclassical bistability for a low-photon cavity occupation has already been experimentally ascertained in \cite{remnants} (see e.g. Figs. 2 and 3 therein), while the discrete nature of the JC spectrum underlies the stability of the two metastable states (vacuum and excited) on approaching the thermodynamic limit, as demonstrated in \cite{StabilityFSC} for large photon numbers.

Once more, spontaneous emission has a drastic impact on the system response. The average photon number in the steady state drops from the value $\braket{n}_{\rm ss}\approx 36.4, 42.8$ to $\braket{n}_{\rm ss}\approx 2.2, 3.7$, for $n_{\rm wc} \sim 10^{-5}, 10^{-9}$, respectively, as we see when comparing frames (c,d) of Panel II to frames (a,b) of Panel III in Fig. \ref{fig:channel}. The excited state is only rarely visited in the course of quantum-fluctuation switching, while the boundary of a first-order dissipative quantum phase transition cannot be crossed for the detuning and drive strength under consideration.  The excited state remains in place in the phase space for varying $\gamma/\kappa$ at a given point in the drive parameter space; what changes is its participation in the cavity output. This is in contrast to what happens in photon blockade, a regime where the entire landscape of quantum dynamics is very prone to decoherence, as we have seen in Sec. \ref{sec:spontBl}.

\section{Symmetry breaking and critical behavior}
\label{sec:symmbr}

The two coherent interactions compete with each other much more clearly on resonance (see also the opening paragraphs in Sec. 16.3.2 of \citep{QO2}), where the neoclassical equations of motion yield a steady-state solution with the typical bifurcation of a second-order phase transition \citep{spontdressedstate, QO2}. Quantum fluctuations organize attractors about the complex-conjugate neoclassical states and realize phase bistability. Above threshold, with $\varepsilon_{d}/g \geq 1/2$, the complex amplitude of the intracavity field is given by the pair of states \cite{spontdressedstate}
\begin{equation}\label{abovethrneocl}
\alpha_{\rm ss}=(\varepsilon_d/\kappa)\{1 - [g/(2\varepsilon_d)]^2\} \pm i\,[g/(2\kappa)]\sqrt{1-[g/(2\varepsilon_d)]^2},
\end{equation}
which shows that cavity occupation grows without bounds as $g/\kappa \to \infty$, following the law (from Eq. \eqref{neocl} on resonance \cite{Carmichael2015})
\begin{equation}\label{abovethr}
|\alpha_{\rm ss}|^2=[g^2/(4\kappa^2)]\,[(2\varepsilon_{d}/g)^2-1],
\end{equation}
something we have already met as an asymptotic form for an intensity-dependent detuning approaching zero with increasing intracavity excitation. The neoclassical equations predict a zero intracavity amplitude at the critical point. In fact, quantum fluctuations produce once more a visible deviation from the mean-field prediction, as we can see in Fig. \ref{fig:BelowthrPB}; on approaching the strong-coupling limit, a bimodal distribution with a long stretch in the phase space emerges from the vacuum state below threshold \textemdash{corroborated} by the switching trajectory drawn in frame (d) of Panel I for $n_{\rm sc}=2.5\times 10^3$, with $\braket{n}_{\rm ss} \approx 29$ [see also the Wigner distribution of Fig. 16.12(c) in \cite{QO2}, at the critical point for $n_{\rm sc}=25$]. In Panel II, we track the development of quantum-fluctuation phase bimodality for growing $\varepsilon_d/g$ below threshold, a precursor to the decay of the unstable vacuum state at a mean-field bifurcation. The two pictures are coming closer together for increasing $\varepsilon_d/g$ above threshold \footnote{We note that in the limit of weak excitation within a dressed-state formulation, where $\varepsilon_d/g \ll 1$, the photon number average differs from zero by $\braket{a^{\dagger}a}_{\rm ss} \approx 2 (\varepsilon_d/g)^4$ \textemdash{see} Eq. (16.203) of \cite{QO2}.}.
\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth]{BistbelowThr.png}
\includegraphics[width=0.5\textwidth]{QbeforeII.png}
\end{center}
\caption{{\it Quantum phase bimodality below threshold.} \underline{Panel I}: {\it Quasi}-probability distributions $Q(x+iy)$ of the intracavity field, obtained by the steady-state solution of the ME, for a fixed drive to coupling strength ratio $\varepsilon_d/g=0.495$, a fixed detuning to coupling strength ratio $\Delta\omega/g=0$ (resonance), and a varying ratio $g/\kappa=5, 10, 100$ in frames {\bf (a)}-{\bf (c)}, respectively. In frame {\bf (d)} we plot a sample trajectory depicting $y(t)={\rm Im}[\braket{a(t)}]$ for the same parameters as in frame (c). The two horizontal lines mark the imaginary part of the two positions in the phase space where the $Q$ function of frame (c) peaks. \underline{Panel II}: Steady-state $Q$ functions at resonance for $g/\kappa=100$ and an increasing ratio $\varepsilon_d/g=0.4, 0.46, 0.48, 0.498$ in frames {\bf (a)}-{\bf (d)}, respectively.}
\label{fig:BelowthrPB}
\end{figure}
\begin{figure*}
\begin{center}
\includegraphics[width=0.5\textwidth]{Bloch.png}
\includegraphics[width=0.48\textwidth]{Fig_Cav.png}
\end{center}
\caption{{\it Symmetry breaking in the quantum picture.} Time-dependent averages from single quantum trajectories with variable driving to coupling strength ratio, $\varepsilon_{d}(t)/g$, ranging from zero to $0.6$ in a linear fashion, and backwards. The overall (in both directions) dimensionless scan time is $\kappa T=1.6 \times 10^3$. {\bf (a)} Bloch sphere representation of the two-level polarization averages $\braket{\sigma_{i}(t)}$ ($i=x, y, z$), with $(X(t), Y(t), Z(t))\equiv (\braket{\sigma_{x}(t)}, \braket{\sigma_{y}(t)}, \braket{\sigma_{z}(t)})$ and initial condition $(X(0), Y(0), Z(0))=(0,0,-1)$ for $g/\kappa=12.5$. The curves in black depict $\braket{\boldsymbol{\sigma}(\varepsilon_{d}/g)}$, calculated from the zero quasi-energy states (see Fig. 2 of \cite{DiracJG2018}), and the red diamond marks the critical point. The inset A shows the time evolution of $Z(t)\equiv \braket{\sigma_z(t)}$ while the inset B shows $y(t)\equiv{\rm Im}[\braket{a(t)}]$. The vertical lines in both insets demark the region above threshold [where $0.5< \varepsilon_{d}(t)/g \leq 0.6$]. {\bf (b)} Phase-space representation of the intracavity amplitude $\braket{a(t)}\equiv x(t)+iy(t)$ for two different values of $g/\kappa$: $12.5$, in blue dots; $25$, in orange crosses. The latter distribution, which corresponds to the superimposed contour plot of the $Q$ function for $\varepsilon_d/g=0.6$, is shown inverted with respect to the center of co-ordinates for clarity of presentation. Solid lines depict the corresponding neoclassical results above threshold from Eq. \eqref{abovethrneocl}. The bold dashed curves depict the two neoclassical branches for $g/\kappa=200$ with $0.5 \leq \varepsilon_{d}/g \leq 0.5 \sqrt{100/99}$, i.e., a scaled drive amplitude reaching only $0.5\%$ above threshold.}
\label{fig:phasebist}
\end{figure*}

In \cite{DiracJG2018}, we read that ``without dissipation, once the system is driven above the critical point, the field radiated by the two-level system is no longer strong enough to interfere destructively with the coherent drive.'' We build upon this remark in Fig. \ref{fig:phasebist}, where the drive amplitude is scanned from zero to a value above threshold and then back to zero in the course of individual realizations for two different values of $n_{\rm sc}$. We first visualize the organizing point and the onset of the second-order dissipative quantum phase transition in frame (a). During the stochastic evolution, the Bloch vector \textemdash{positioned} initially at the south pole of the sphere \textemdash{fluctuates} about the neoclassical curve defined by $X(t)\equiv\braket{\sigma_{x}(t)}=-2\varepsilon_{d}(t)/g$ and $Z(t)\equiv\braket{\sigma_{z}(t)}=-\sqrt{1-[2\varepsilon_{d}(t)/g]^2}$ and, after meeting the critical point, follows two separating symmetric distributions positioned about the equatorial plane above threshold, as inset A shows.

The sample trajectory in frame (a) of Fig. \ref{fig:phasebist} follows closely the average polarization $\braket{\boldsymbol{\sigma}(\varepsilon_{d}/g)}\equiv\braket{\phi_0|\boldsymbol{\sigma}|\phi_0}$ in the zero quasi-energy eigenstate $\ket{\phi_0}$ (with dissipation {\it a priori} absent), which captures the symmetry breaking in alignment with the prediction of the neoclassical theory in the presence of (driving and) dissipation \cite{DiracJG2018}. We also note that fluctuations intensify significantly prior to attaining the critical point, in contrast to the response of the $\sqrt{n}$ anharmonic oscillator, as we can observe in the imaginary part of the coherent-state amplitude of the resonant cavity mode, drawn in inset B. The imaginary part of the intracavity amplitude fluctuates weakly in the beginning about the neoclassical state $\alpha_{\rm ss}=0$ \cite{spontdressedstate} before alternately switching between the two attractors determined eventually by Eq. \eqref{abovethrneocl} past the critical point and beyond the region defined by the critical behavior we met in Fig. \ref{fig:BelowthrPB}. These mean-field attractors are the states drawn by the solid lines in the phase-space distribution of frame (b). Their deviation with respect to the $x$-axis in the phase space is a monotonically decreasing function of $\varepsilon_d/g$, with an angle
\begin{equation}\label{thetass}
\theta_{\pm}= \pm \arctan\{[(2\varepsilon_d/g)^2-1]^{-1/2}\}.
\end{equation}
This expression gives $\theta_{\pm} \approx \pm g/(2\varepsilon_d)$ in the limit $2\varepsilon_d/g \gg 1$. In the same limit, the Bloch vector fluctuates maximally between the attractors $(X=0, Y=\pm 1)$ on the equatorial plane.

When quantum fluctuations are taken into account, the separation between the two distributions of phase bimodality in the phase space of the cavity field decreases when lowering $n_{\rm sc}$; in the extreme case where $g/\kappa \approx 1$, they merge to produce eventually a single squeezed coherent state in the steady-state response. In this case, squeezing of fluctuations occurs along the real axis, which is the direction of the neoclassical excitation amplitude $\varepsilon_d/\kappa$; this amplitude need not be large.

Above threshold, quantum fluctuations are positioned around the neoclassical curves, more clearly so for growing $\varepsilon_d/g$. The discrete quasi-energy spectrum has now collapsed and is determined solely by the continuous variation of the drive amplitude \cite{QO2}. Hence, with increasing the value of $n_{\rm sc}$ we attain higher and higher cavity photon numbers for the same ratio of $\varepsilon_d/g$, scaling once more as $|\alpha_{\rm ss}|^2=(\varepsilon_{d}/\kappa)^2$ for $2\varepsilon_d/g \gg 1$, as predicted by Eq. \eqref{abovethr}. The coupling strength $g$ is manifestly absent in this limit, testifying to the vanishing interference with the field re-radiated by the two-level atom \textemdash{a} sharp contrast with the formation of multi-photon resonances in the region of photon blockade.

On approaching the thermodynamic limit, the fluctuations follow suit to stabilize the mean-field states of phase bimodality against the neoclassical prediction of nonstable limit cycles above threshold \cite{spontdressedstate}, as we can see in frame (b) of Fig. \ref{fig:phasebist}. Moreover, for $2\varepsilon_d/g \gg 1$, deviations due to quantum fluctuations in the $Q$ representation from the attractors comprising the bimodal distribution in the steady state are constrained inside a ring delimited by a coherent-state uncertainty, as opposed to the replication of a bifucation in the phase space that we met in Fig. \ref{fig:BelowthrPB} when accessing the region below the critical point. Those fluctuations also appear symmetric in the phase space and are squeezed in the radial direction, with the radius itself defined by the large coherent-state amplitude $\varepsilon_d/\kappa$. Extending finally to the generalized Jaynes-Cummings-Rabi model, we note that the many-photon strong-coupling limit obviates the need of employing many two-level emitters to attain high excitation; strong coupling to one emitter is sufficient \cite{QPTJCRabi}.

\section{Concluding discussion}

In conclusion, we have explored the definition of a strong-coupling thermodynamic limit for the driven dissipative JC interaction. This limit probes the ability of the field radiated by the two-level system, the fundamental element introducing the nonlinearity, to interact coherently with a resonant mode driven by a fixed, externally-imposed drive, when at the same time this coherent interaction is compromised by dissipation. We have shown that a strong coupling between the atom and its radiated field, resonating in the cavity, underlies the manifestation of photon blockade in the parameter regime where $\varepsilon_d \ll g \sim |\Delta\omega|$. Distinct resonance peaks mark the formation of effective two-level complexes with varying participation in the response as we attain the strong-coupling limit. The signal leaving the cavity comprises the individual photons taking part in this coherent process rather than the classical drive field, detuned off-resonance, despite the fact that the scaled semiclassical amplitude vanishes. As we access the region $\varepsilon_d \sim g \gg |\Delta\omega|$, we have demonstrated that the drive and re-radiated fields are progressively placed on equal footing, shaping the picture of bimodality in the phase space. Instead of the individual multi-photon resonances there is now a split-Lorentzian response \textemdash{evidence} of the ``collective'' form the nonlinearity now assumes \textemdash{with} peaks at detunings approaching a zero value for vanishing dissipation, which is where symmetry breaking takes place.

Moving now to resonance, $\Delta\omega=0$, below threshold the two fields interfere destructively to produce a zero neoclassical intracavity amplitude. On the other end, when driving above threshold with $2\varepsilon_d/g \gg 1$, the parameter $n_{\rm sc}$, scaling the cavity excitation to very large values, accounts only for a correction to the mean-field cavity output with respect to the empty-cavity response. In the region of the critical point, however, we have demonstrated the presence of significant quantum fluctuations against the neoclassical response, heralding symmetry breaking and the associated bifurcation; the ME predicts a bimodal steady-state distribution below threshold with an appreciable photon-number average in the strong-coupling limit. For high excitation above threshold, we can invoke the two excitation ladders of the JC spectrum, although in this case separately driven and with a square root of the corresponding bosonic number operator replacing the atom-field coupling term in the JC Hamiltonian. Fluctuations induce switching between these two ladders, conforming to mean-field phase bistability.

In fact, bistability provides the connective net between the aforementioned frames, some times rather loose and others more palpable; we encounter it from the weak semiclassical deflection of the vacuum Rabi resonance up to the phase-bistable switching at resonance involving potentially innumerable photons. We have demonstrated, however, that quantum fluctuations are not there to merely realize of forecast mean-field bistability: the approach of a strong-coupling thermodynamic limit in photon blockade entails uncompromised resonances in continuing disagreement between the quantum picture and the semiclassical treatment. As we have also observed, among the two decoherence channels, cavity decay is not as effective in inducing switching between the JC excitation ladders as spontaneous emission, which justifies the special place reserved for the neoclassical equations and the pertinent scaling law.

With the above remarks and qualifications in mind, sending the strong-coupling scale parameter $n_{\rm sc}$ to infinity exposes the underlying JC energy spectrum in either its discrete or continuous manifestation, depending on the relative weight of the two coherent coupling strengths. Quantum fluctuations determine the particulars of critical behavior, as captured by the experiment, based on the interplay between the coherent and incoherent interactions.

\begin{acknowledgments}
I am grateful to H. J. Carmichael for his unfaltering guidance, and to J. Larson for several inspiring discussions. This work was supported by the Swedish Research Council (VR) as well as by the Knut and Alice Wallenberg foundation (KAW).
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
King of the Midsize Trucks
2011, Toyota, Tacoma
When you've got the best-selling compact pickup in the U.S. for the last six years, you don't need to do a whole lot to keep things rolling. So, the Tacoma gets minor changes this year, such as a new grille and standard air conditioning on all models. Toyota also is introducing a couple of new packages based on a concept truck from the 2009 SEMA show. And, with environmental and fuel cost concerns on folks' minds, there are more four-cylinder models now. The Tacoma offers an array of configurations. Do you want a Regular Cab (two doors) or Double Cab (full four doors) or the Access Cab that splits the difference? Long or short bed? Two- or four-wheel drive? Four or six cylinder power? And, which of the numerous option packages would you like to add? That's a huge range, from the regular cab, four-cylinder, two-wheel-drive worker bee to the long, loaded double-cab model with V6, four-wheel drive and long bed-like my Speedway Blue Metallic tester. Vehicle curb weights vary from 3,250 pounds to 4,190 based on these choices. My tester featured the 4.0-liter, dual-overhead-cam 236-horsepower V6, with 266 lb.-ft of torque. The standard 2.7-liter four-cylinder, available in five models, generates 159 horsepower and 180 lb.-ft. of torque. Both engines use Toyota's variable valve timing for maximum power at a wide range or rpm. The V6 offers a six-speed manual transmission or five-speed automatic; the four-cylinder comes with one cog less-five speed manual or four-speed automatic. The EPA gave my V6 model ratings of 16 City, 20 Highway (18 average). The Green Vehicle numbers are 5 for Air Pollution, 3 for Greenhouse Gas. The base 4-cylinder truck with manual earns 21 City, 25 Highway (and 5/5 on the Green scores). There's something cool about riding high and strong along America's highways-it's kind of infectious. The Tacoma's interior is handsome, as is the outside, but there's little padding there, so you feel tough already. Having that bed behind you is an invitation to do some heavy lifting. The standard payload is about 1,300 pounds. My truck had the optional 73.5" long bed, which is standard on most models. However, on Double Cabs, the shorter five-foot bed is standard, and it might even be desirable with the already stretched platform for parking considerations. Both Tacoma beds are lined with a composite of tough sheet-molded compound (SMC). It's ten percent lighter than steel and won't rust. The beds feature two-tier loading and an integrated deck rail utility and four tie-downs. My truck had the optional fold-out bed extender, which lets you carry cargo with the tailgate down. On the Standard Cab, Access Cab and Double Cab models, you can order a PreRunner version, which has the look and feel of the four-wheel-drive model but with two-wheel drive underneath. The X-Runner is a sport model. It features the "X-Brace tuned suspension," special 18-inch wheels and tires, and a striking body kit. It comes only as an Access Cab and with a manual transmission. You can go from zero to 60 in less than 7 seconds, and it really sticks to the road with lateral acceleration scores (more than 0.9g) in the vicinity of some sports cars. Toyota's new specialty models this year include the T|X and T|X pro, which use the special offroad package from Toyota Racing Development (TRD) to make some eye-catching and potent vehicles. The T|X adds black alloy wheels, rugged-looking tires, black tube side steps, a stainless steel exhaust tip and special exterior graphics. The T|X Pro has is the higher performance version with a more thrilling engine roar. TRD also offers a supercharger for any V6 model, which brings the engine up to 304 horsepower and 334 lb.-ft. of torque. The Tacoma is no "mini-truck" like the first small Japanese pickups. The trucks have either a 109.6 or 127.4-inch wheelbase and can stretch to as much as 221.3 inches (nearly 19 feet) long. At a full six-feet wide, the cabin is roomy-and I could even (barely) accommodate an upright bass in the passenger seat of my Double Cab. You can haul from 3,500 to 6,500 pounds, depending on your model and presence of the towing package. Prices start at $16,365 for the Standard Cab four-cylinder with two-wheel drive and work up from there. My top-of-the line model listed at $27,525; when you added in the SR5 Extra Value Package (lots and lots of welcome extras), the V6 Tow Package, and things like the Hood Protector ($119), it came to $32,704. Between the different body configurations, two engine and transmission selections, a four-wheel-drive option, numerous equipment packages, colors and long list of standalone options, you can pretty much spec out exactly the Tacoma you desire.
Images of the 2011, Toyota Tacoma
Tacoma gets minor changes this year, such as a new grille and standard air conditioning on all models
The Tacoma's interior is handsome, as is the outside
For 2011, Tacoma's standard payload is about 1,300 pounds
The Tacoma is no "mini-truck" like the first small Japanese pickups

redpajama_set_name: RedPajamaCommonCrawl


text:
Home Improvement's Zachery Ty Bryan arrested for 'strangling woman'
© Provided by Daily Mail
Zachery Ty Bryan, 39, was arrested late on Friday in Eugene, Oregon, after a neighbor called to report a domestic dispute
Actor Zachery Ty Bryan, who played the oldest son on the 1990s sitcom Home Improvement, has been arrested after allegedly strangling his girlfriend.
Bryan, 39, was arrested late on Friday in Eugene, Oregon, after a neighbor called to report a domestic dispute.
According to a police report, Bryan's girlfriend said the two of them were arguing when he wrapped his hands around her neck and squeezed, TMZ reported.
The girlfriend said that when she tried to call 911, Bryan tried to grab her phone away from her, and that she fled across the street to hide.
Arriving on the scene, cops found Bryan sitting outside his home and the girlfriend at the neighbor's house.
Bryan was charged with strangulation, a felony, and misdemeanor charges of fourth-degree assault and interfering with making a report.
Bryan (second from left) played the oldest son on the 1990s sitcom Home Improvement
On Saturday afternoon, Bryan remained in custody at the Lane County Jail, and it was not immediately clear if he had an attorney to speak on his behalf.
MailOnline logo
Bryan was a main cast member of Home Improvement for the series eight-season run from 1991 to 1999.
He portrayed Brad Taylor, the eldest son of Tim Taylor (Tim Allen), even though he was one month younger than Jonathan Taylor Thomas, who played his younger brother Randy.
His character was known for experimenting with different hair styles, as well as being the child who most frequently got in trouble.
Developing story, check back for updates.
Inarrested, Bryan, Home, Improvements, strangling, Woman, Zachery
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As We Remember - email spam message. This message appears sent from - debbie@newsletter.militaryconnection.com.
In our last newsletter, we asked our readers to share their memories of where they were when they heard about the 9/11 attacks. We would like to thank all of you who have sent in your submissions, and we are featuring a number of them on our blog . We hope that on Sunday, you will take the opportunity to honor the memories of those who perished by flying the flag and participating in an event in your area that celebrates our freedoms. Although we may have different opinions and political views, 9/11 is a day to come together and remember that we live in the greatest nation in the world. Let's remember those who sacrifice to preserve these freedoms we hold dear.
The Armed Forces Vacation Club is a "space available" program that offers military and other Department of Defense-affiliated personnel the opportunity to enjoy vacations at popular destinations around the world – for the incredibly low price of just $349* USD per unit, per week.
Those eligible and their family and friends have access to spacious accommodations at more than 3,500 resorts, apartments, condominiums and homes in more than 80 countries. Stay with your loved ones at a beachfront resort in Mexico, a well-appointed cabin in the Canadian Rockies, or a centuries old home in the English countryside. Take your leave at a do-everything resort near Orlando's theme parks, Greece's ancient ruins or Tuscany's rolling hills.
With Armed Forces Vacation Club, your vacation options span the globe.
Employer Support of the Guard and Reserve is a Department of Defense agency established in 1972 to develop and maintain employer support for Guard and Reserve service. ESGR advocates relevant initiatives, recognizes outstanding support, increases awareness of applicable laws, and resolves conflict between service members and employers. Paramount to ESGR's mission is encouraging the employment of Guardsmen and Reservists who bring integrity, global perspective and proven leadership to the civilian workforce.
MilitaryConnection.com is proud to once again join forces with the Ronald Reagan Library to remember the victims of September 11th and honor our troops and first responders. The Annual 9/11 Freedom Walk Program is one of thousands that will be held nationwide to mark the 10th anniversary of the attacks against our nation. Let us memorialize those who died and thank those who serve in the military, law enforcement, first responders and medical professionals.
Homefront America Needs Your Vote!
Tech EXPO Explore new career opportunities this September! TECHEXPO Top Secret brings Security-Cleared professionals together with the nation's leading employers from the defense & government sectors. Interview for the many opportunities available in cyber security, information technology, engineering, aerospace, telecom, project management, intelligence, operations, homeland security, research & more at TECHEXPO Top Secret's September hiring events! You may also submit a resume online & apply to thousands of available positions!
Our Military Kids wants to help your family through your deployment. If you have a child age 3 to 18 with a parent currently deployed overseas with the National Guard or Reserves, your child is eligible for a grant. The grants, which can be up to $500, will pay for up to six months of an extracurricular activity of your choice. The process is simple. Visit their website and select the appropriate one-page application . Once the application is complete, send it with supporting documentation. That's it. If you have any questions, you can visit their FAQ page or call (703) 734-6654. They can only provide grants to children with a parent who is currently deployed – if your service member has yet to deploy, or has already returned home, unfortunately, the child is not eligible.
VAMBOA is a non-profit trade association working to ensure the development, growth and prosperity of Veteran Owned Businesses, Service Disabled Veteran Owned Businesses (SDVOB) and Military Businesses of all sizes. VAMBOA will serve the needs of the veteran and military business communities by uniting veteran and military business owners for collaboration, connections and contracts with both private corporations and government agencies. Join today and take the survey to let them know what issues are important to you. Membership is Free ! You have honorably served our nation. VAMBOA is here to serve you and be your voice.
We have reached a milestone on Twitter…over 88,000 followers! For lots of interesting tidbits, follow us at http://twitter.com/thegotosite .


Each of these services and the associated costs are explained below.
An initial meeting with one of our qualified financial planners.
A chance to ask questions and understand what we do.
An opportunity for us to find out what help you require.
A discussion of the options available to you from our menu of services.
A refinement of your financial goals.
A background knowledge of your current financial circumstances and personal information.
An investigation into your knowledge and experience with investments.
An assessment of your attitude, tolerance and capacity for investment risk.
We read all the small print on your existing financial planning.
We use industry leading technology to source the most suitable products and funds.
We collaborate with our colleagues who are specialists in their respective fields of finance.
We consider all the various tax strategies .
We decide on an asset allocation which matches your risk profile.
We draft your financial plan.
A comprehensive understanding of your current financial planning and how close to achieving your financial goals you are.
An understanding of our advice and recommendations to improve you financial planning where needed.
An opportunity to ask any questions which arise.
An opportunity to fine tune any details.
When you receive our report/financial plan it will be your decision whether to implement our recommendations either completely or in part.
A clear breakdown of what our fees are for implementations is included in our client agreement which you are in receipt of.
The handling of all fund and policy administration on your behalf.
Regular updates to keep you informed of progress.
Ensure all your documents are issued in line with your expectations.
Confirmation of all actions taken on your behalf in writing.
Structured reviews to give you piece of mind.
Assessment of your current circumstances and any changes to your plans that are needed.
Regular updates and information regarding your holdings.
Ongoing support with correspondence and administration issues.


Need a Commercial Dumpster Service in Littleton, NH?
It is likely you have ordered the fresh new rug and it is bundled up and the safeguarding plastic will have to be thrown out in the dumpster. Once the sheetrock is completed you are able to install the brand new rug or have carpeting layers complete the task for you.
Initial, determine what measurement dumpster you need to assist remove the waste material and debris left from your renovations. Placing all debris in the dumpster often will keep the renovation web page clean and pay off the entire time. Brilliance is seen in terms you cope with the site with the renovation and the end results. You may display a project that is a lovely demonstration of brilliance by having a dumpster shipped right on web page from the beginning with the project. Most subcontractors must be asked to dump their waste items in the can each evening before leaving the website and extra supplies should be given a location on the site for a in the future project or to be given.
The companies grab the garbage as well as recycle that which makes you have rid of the garbage in a natural way. These lenders provide containers according to the rubbish quantity. This kind of bins can be simply seen with construction sites. They come in various shapes and sizes to carry garbage as a result. So in line with your celebration trash, you are able to select the correct bin that may carry various quantities of rubbish.
Dumpster diving is an easy way to get additional money. It is like any treasure hunt but you are surrounded by rubbish. You can easily come across valuable objects and sell these without any cost to you. Examine your local regulations and give that a shot.
Methods include: Evaluate which size dumpster you may need. Do you need a full sized or possibly a small dumpster. Take into account the amount of junk that will have to be removed by the completion of the project. Determine if you want a commercial or residential rental. An industrial rental may be needed if your contractor is progressing a huge remodelling and is getting rid of bricks as well as wood etcetera. Online online catalogs provide every one of the answers you may need for selecting the right size and elegance. If you live in the Washington DC area you can need a Maryland Commercial Dumpster Service in Littleton, NH; you can even go online and place an order as well as call. A rapid removal of the full dumpster is available. The construction procedure can be made less complicated with a Commercial Dumpster Service in Littleton, NH.
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Mirror for Samsung TV 3.5 macOS .
Mirror for Samsung TVworks on any Samsung Smart TV from the 2012 models onward.Which model year do you have? You can see that by looking at the middle letter in your model type (on the back of your Samsung TV).
Mirror for Roku 2.7.1 macOS .


text:
Carpino of Connecticut, Louis of New Mexico to Join NCLGS Board
LAS VEGAS – Sen. Keith Pickard, president of the National Council of Legislators from Gaming States (NCLGS), announces a new leadership slate for 2021 to advance the critical goal of promoting best practices in gaming policy.
Two new board members are:
Connecticut State Representative Christie Carpino was first elected to the Connecticut Legislature in 2010 and was recently re-elected for her 6th term. She is the House Republican Caucus Chair in the Connecticut General Assembly, and serves on the Government Administration and Elections, Public Health, Legislative Management and Regulations Review Committees.
Representative Georgene Louis of New Mexico chairs the New Mexico House State Government, Elections & Indian Affairs Committee and serves on the House Judiciary and Rules & Order of Business Committees. First elected in 2012, she is the first Pueblo woman to serve in the New Mexico Legislature, and would be the first Native American lawmaker to serve as an Officer of NCLGS.
Additionally, Florida Senator Perry E. Thurston Jr., who has long served as Treasurer of NCLGS, will become the organization's Vice President.
Other state legislators who will continue to serve as officers are:
Senator Jon Ford of Indiana
Delegate Shawn Fluharty of West Virginia
Former Sen. William Coley of Ohio, who served for two years as president of NCLGS, and Rep. Brandt Iden of Michigan, who served as an officer of NCLGS, have been term-limited in their respective states and will no longer serve as officers.
"Bill Coley and Brandt Iden served with distinction as leaders in their states and as reasoned voices on the national stage to promote best practices in gaming legislation and regulation. All of us thank them for their hard work and successful efforts," Pickard said. "Christie Carpino and Georgene Louis have already made their view on gaming known to us at NCLGS, and we know them to be thoughtful leaders who will work hard on a national level to ensure legal gaming reaches its potential throughout the United States."
The new slate of officers will be sworn in during a virtual NCLGS session to be scheduled later this month, and will assume leadership roles at the NCLGS 2021 Summer Meeting, July 12-13 2021, to be held at the InterContinental Magnificent Mile in Chicago.
About NCLGS
NCLGS Is the only organization of state lawmakers that meets regularly to discuss issues relating to gaming. Members of NCLGS serve as chairpersons or members of state legislative committees responsible for the regulation of gaming in their state legislative houses. NCLGS does not promote or oppose gaming but is primarily concerned with the regulation and economic and social impacts of the industry. The NCLGS Foundation is the educational and research arm of NCLGS. The 501(c)(3) non-profit is a source of non-partisan data on issues of gaming legislation and regulation.
About Spectrum Gaming Group
Spectrum Gaming Group serves as Executive Director of NCLGS.Spectrum has consulted for gaming operators, regulators, and legislatures in 40 US states and territories and in 48 countries on six continents. Spectrum's clients include 22 US state and territory governments, six national governments, 22 Native American entities, numerous gaming companies (national and international) of all sizes, suppliers, lotteries, financial institutions, developers, and other gaming-related entities.


text:
Putin Offers Condolences in Wake of Charlie Hebdo Shootings in Paris
Russian President Vladimir Putin has expressed his condolences to those affected by an attack on the offices of French satirical newspaper Charlie Hebdo on Wednesday that claimed at least 12 lives.
Twelve people, including two police officers, were killed Wednesday when black-hooded gunmen stormed the Paris offices of the satirical weekly.
During the attack, the gunmen shouted "we have avenged the Prophet Muhammad," witnesses told the BBC.
The weekly was known for its irreverent caricatures of politicians and religious leaders.
Presidential spokesman Dmitry Peskov told the TASS news agency that Putin offered his condolences to all the victims' loved ones, as well as all Parisians, and stressed that the Kremlin condemns terrorism in all its forms.
French President Francois Hollande said he has no doubts Wednesday's incident was a terrorist attack, according to Reuters. It was unclear how many attackers there were and whether they had been apprehended, but Reuters cited a witness as saying several men had entered the building with automatic rifles.
Charlie Hebdo's offices were firebombed in 2011 after publishing cartoons featuring the Prophet Muhammad.
Islamic State Suspected in Chechnya Terrorist Attack
A new breed of Caucasus extremists is behind the terrorist strike in the Chechen capital Grozny — and they just may be connected to the terrorist...
KLEPTOCRACY
In list drawn up before his detention, Alexei Navalny named Roman Abramovich as top target for Western sanctions, according to key ally.


The principal and one teacher from Anuban kindergarten school arrived today, happy but sleepy from their 18-hour bus trip from Muang Khong. We held a pre-training meeting where we went over the schedule for tomorrow, and also discussed the progress of the improvements at the school. They were extremely happy about all the changes and said the community and the students are all looking forward to the completion of the project. We were very impressed with the level of leadership and accountability they have shown on this project--they and the other teachers did some of the work themselves (such as painting) and tried to get the community involved as well. They also made sure to keep the costs as low as possible, accounting for everything.
We still have many things to work on together--such as creating a fence around the school to protect the small children and allow them to grow fruit trees and other things (right now the wandering water buffalo would eat them!). Tomorrow we will be leaving at 6:30 am to go to the Sunshine School. We may have a meeting on Tuesday with another person running a charity here, but will have to see how the timing goes. For now...must get to sleep! It's a full day tomorrow!


text:
Q: Change DCA amount on a strategy depending on how much you spent last month I am developing a strategy for Bitcoin where you DCA every month. You specify de maximum quantity you are willing to DCA like so:
DCAamount   = input.int(1000,"DCA Amount",400,tooltip="Monthly DCA")

Furthermore, the amount you will actually DCA from the specified maximum (1000) is modified depending on the current risk in the market, calculated from 1-10. If the risk is the lowest you will buy the maximum, if it is higher you will buy lower amounts and if it is above 5 you will sell. The higher the risk the more you sell.
Currently I specify it like so:
y       = DCAamount / 5
switch
    band10 => y:= y * 100
    band9 => y:= y * 20
    band8 => y:= y * 10
    band7 => y:= y * 5
    band6 => y:= y * 2
    band5 => y:= y * 1
    band4 => y:= y * 2
    band3 => y:= y * 3
    band2 => y:= y * 4
    band1 => y:= y * 5

if testPeriod()
// Buy orders
    if T_buy == true
        if dayofmonth == day
            strategy.entry("Risk Band",strategy.long,qty=contracts,when=band5)
            strategy.entry("Risk Band",strategy.long,qty=contracts,when=band4)
            strategy.entry("Risk Band",strategy.long,qty=contracts,when=band3)
            strategy.entry("Risk Band",strategy.long,qty=contracts,when=band2)
            strategy.entry("Risk Band",strategy.long,qty=contracts,when=band1)

// Sell orders
    if T_sell == true
        if dayofmonth == daysell
            if strategy.opentrades > 1
                strategy.close("Risk Band",qty=contracts,when=band6)
                strategy.close("Risk Band",qty=contracts,when=band7)
                strategy.close("Risk Band",qty=contracts,when=band8)
                strategy.close("Risk Band",qty=contracts,when=band9)
                strategy.close("Risk Band",qty=contracts,when=band10)

I would like to improve this strategy by adding the amount of DCA money left each month to the next one: Let's say the risk was 2, with a maximum DCA of 1000 and using the code above that month we would buy 800$ of BTC. I would like to include this "extra" 200 to the calculation of the DCA for the next month.
Also, in months where risk is >5 I would not be buying any BTC but selling , but I would still like to add 1000$ to my stack for investing in the months where the risk finally comes down to lower levels.
I was thinking I should create a variable that adds the remaining amount I didn't spend and uses it to calculate next month buy, but I'm struggling with that.
Thanks

redpajama_set_name: RedPajamaStackExchange


Ed Ou is a staff visual journalist and documentary filmmaker with NBC News.
He started his career early as a teenager, covering the 2006 war between Israel and Hezbollah in southern Lebanon, and the fall of the Islamic Courts in Mogadishu, Somalia while he was studying in the Middle East. He first worked for Reuters and the Associated Press, covering a wide range of news stories in the region. He then worked for the New York Times covering East Africa and the Middle East during the 2011 Arab Spring uprisings. Since then, he has covered indigenous stories in Canada, the drug war in the Philippines and global warming in Europe for NBC News.
Ed has been the recipient of a Global Vision Award, World Understanding Award, Documentary Journalism Award, and Photographer of the Year Award from POYi, a 1st Place Contemporary Issues award from World Press Photo. He has also received an International Reporting Award from the Overseas Press Club, and other recognitions from the Ian Parry Scholarship, Best of Photojournalism, PDN Photo Annual, UNICEF, among others.
He has been selected for a Getty Images Editorial Grant, PDN 30 Under 30, and took part in the World Press Photo Joop Swart Masterclass. He was also awarded the City of Perpignan Young Reporter Award at Visa Pour L'Image and the Young Reporter Prize from the Prix Bayeux-Calvados Award for War Correspondents.
He is a TED Senior Fellow.


A bombshell for Arctic Monkeys fans has happened, it seems that the band will not be releasing any singles before the 11th of May, the date when the album is out. A Domino representative has confirmed to Pitchfork that fans won't be able to hear any new songs from the LP until its official release, which means it could be a concept album for example.
Still, not long to wait for their brand new album! Which almost became an Alex Turner solo record, while speaking to MOJO guitarist Jamie Cook said "I think at first, because it was quite basic – piano, vocal and no guitar – Al was in two minds about, 'is this Arctic Monkeys or am I going somewhere else with this'. And maybe at first I was a bit like that as well. It's not definitely not a guitar-heavy record, not typically what we'd do. It took a lot more thinking about."


When a patient is diagnosed with MPS I or MPS II disease, knowledge and access to information are powerful tools.
Patients who are educated about their condition and treatment options are better prepared to manage their own health, engage in important dialogues with healthcare professionals and the general public.
To assist with this process we have provided you with links to local and international support networks and patient support programmes that will help patients with their journey.
We are committed to empowering patients through knowledge to ensure they fully understand their condition.
Sanofi Genzyme understands that managing work and family while maintaining such an infusion schedule can be challenging. To assist patients to fit their infusino schedule into their lifestyle, Sanofi Genzyme is proud to introduce the Care for Rare Home Infusion Program.
Patient support groups provide support and information on MPS I and MPS II disease bringing the disease community and others living with genetic disorders closer together.


Window shopping is what women and girls love, passing by brightly lit displays flashing dazzling ensembles in traditional and contemporary designs brings a euphoric feeling similar to that of real love. For shopaholics it can be torture while for the rest it is pure eye candy.
Stylish western wear is always in fashion and very much in demand owing to the unseasonal stay of winters in Karachi. Return of the winter chills in Karachi means one thing- shopping furthermore for winter accessories. Surprisingly this winter chill is welcomed by most while some have further gone in hibernating until the next sunshine.
Winter crazy people are out and about making statements with their attires. This here is time for them to make use of all those bundled up winter wear. Hoodies and jackets have become essential style statements with the gorgeous ankle length boots; a complete winter ensemble for girls. However the men are not far behind in this race; they have adorned themselves in jackets complemented by coloured trousers, adding a fair bit of brightness to the grey winter mornings. Fashion statements are often costly and heavy on the pocket for most of the people; leaving them with the choice to wait for birthdays or winter clearance sales.
Fifth Avenue is a store that is creating ripples since it started, when the concept of locally branded western wear was unknown here in Pakistan. Offering stylish western wear throughout the year for all seasons, it serves as a one stop shop for the young. Shopping here is often restrained for most by the size of the pockets, but the periods of waiting and longing are over.
Emerald Tower, 2 Talwar, Clifton, Karachi.
The Ocean Mall, 2 Talwar, Clifton, Karachi.
Unbelievably true, the store is offering a the grand sale discounts of up to 70% on all the winter products and this is for starters- flat discounts of 50% on jeans for women and men while a 30% flat discount on new arrivals is up for grabs. Astonishing? Yes! Wasting time in wondering if this really true would be a sheer waste of opportunity.
Stylish coloured jeans are in vogue for their eccentric colours which will brighten up any day or dull morning- available for both men and women. Fifth Avenue has revolutionized the styles of jackets- jackets have gone from fat and bulky to slim fit with studs or strapped around with chunky or sleek leather belts.
An ensemble for women is incomplete without a trendy hand bag; the ones available at Fifth Avenue are sure to fit any occasion whether formal or causal. Available in bright colours like blue, yellow and red, these hand bags are the must haves for this winter season. With the cold biting at the toes, not many will opt for open shoes or sandal, instead the ankle or long boots edged with fur will prove to be better foot wear choices- both elegant and confortable.
Excitingly all of these items are available at astonishingly huge discount prices. Fashion statements with Fifth Avenue are the way to roll. Thus without further ado head out and shop at the nearest outlets in Park Towers or Ocean Mall!!


Review Request #130228 - Created Aug. 16, 2017 and submitted March 25, 2018, 1:09 p.m.
Aug. 17, 2017, 8:14 a.m.
Looks OK. Nice code duplication, but ok, this is kdelibs4support, not worth refactoring.
PS: Please use phabricator, we're told reviewboard is on the way out.
Ralf Habacker March 25, 2018, 1:08 p.m.
March 25, 2018, 1:08 p.m.


In 1729 eighteen singing men of the parish clubbed together to build a choir gallery in the parish church of St Lawrence, Gnosall, Staffordshire. The gallery or loft was situated next to the pulpit, rather than at the rear of the church.
That the Loft in the Church of Gnosall by the Pulpit was by the Minister and church-wardens appointed for the use of certain Persons to sing Psalms there.
That it was fitted for that use at the expence of Thomas Fowke, John Stevenson, John Hicken, John Collier, William Collier, Thomas Ward, William Adderley, John Chilton, Humphrey Bayley, Thomas Sutton, Nathaniel Sutton, William Bromley, William Venables, John Parkes, John Lees, Adden Ashton, Richard Bernard, and William Reynolds, and that they are to enjoy the said Loft during their continuance to sing Psalms to demean themselves well.
That the expence in fitting the said Loft for that purpose did amount to the sum of one pound and sixteen shillings.
That four pence a year shall be allowed by every one of the above-named Persons for his sitting in the said Loft till the said sum of one pound and sixteen shillings shall be discharged.
That if any one of those Persons who are appointed to sit there shall leave his place, another Person, who can Sing Psalms shall be nominated by the Minister and Church-wardens of Gnosall to succeed him and that he who is nominated to succeed him shall pay to him, that resigns his place or Sitting, the eighteenth part of that which shall then be unpaid of the one pound and sixteen shillings.
Robert Reynolds and John Alderley were elected into the places of William Reynolds and John Lees by the minister and church-wardens.
Elsewhere, galleries were provided by wealthier donors, such as John Ford at Offwell in Devon (who was patron of the living). The choirs in both parishes would have been singing metrical psalms, perhaps using Nahum Tate and Nicholas Brady's New Version of the Psalms of David (1696), which drew on Thomas Sternhold and John Hopkins' Whole Book of Psalms in English Metre, published in 1562 and still in use over a century later.
Like most Georgian galleries, the one at Gnosall would have been destroyed by the Victorians. The gallery at Offwell, which was at the west end of the church, was built in 1754 and removed exactly a century later. The loft at Gnosall, if it survived until 1820, would then have been swept away in the major rebuilding of the church which took place in that year.