text:
\section{Introduction}\label{sec:sec1}

Temperature is the most fundamental cornerstone in the theory of both classical and quantum thermodynamics. How to measure the temperature of a quantum reservoir with a high sensitivity has recently attracted much attention~\cite{PUGLISI20171,Mehboudi_2019,PhysRevLett.125.080402,PRXQuantum.2.020322,Mok2021,PhysRevResearch.3.013244}. Quantum thermometry pursues a highly precise measuring of temperature, which can surpass the standard bound set by classical statistics, with the help of coherence, quantum squeezing, entanglement, or other quantum resources. In a typical thermometry scheme, a two-level system~\cite{PhysRevA.98.050101,PhysRevA.99.062114,Tamascelli_2020} or a harmonic oscillator~\cite{PhysRevA.96.062103,PhysRevB.98.045101,PhysRevA.103.023303}, working as the quantum sensor, is coupled to the quantum reservoir of interest. Due to the sensor-reservoir interaction, the information about the reservoir's temperature is encoded into the state of the sensor. Then, by measuring a certain sensor's observables, the knowledge about the temperature can be obtained. A highly sensitive quantum thermometer has wide applications in the fields of physics, biology, and material science.

Roughly speaking, there are two totally different quantum thermometers in the present existent temperature sensing schemes: the fully thermalized thermometer~\cite{PRXQuantum.2.020322,PhysRevA.96.062103,PhysRevB.98.045101,PhysRevLett.114.220405,Potts2019fundamentallimits,PhysRevA.102.012204,Latune_2020} and the partly thermalized thermometer~\cite{PhysRevA.98.050101,PhysRevA.99.062114,Tamascelli_2020,PhysRevA.101.032112,PhysRevX.10.011018,PhysRevA.103.012217,PhysRevResearch.3.013244,Kenfack2021,L_pez_V_zquez_2020}. If the sensor-reservoir interaction is weak and the encoding time is sufficiently long, one can regard the sensor as being completely thermalized, namely, the sensor evolves to its thermal equilibrium state which is independent of the encoding time. In many previous treatments~\cite{PRXQuantum.2.020322,PhysRevLett.114.220405,Campbell_2018}, such a thermal equilibrium state is approximately expressed as a canonical Gibbs state at the same temperature as the reservoir. In this case, the temperature can be readout directly from the canonical Gibbs state, and the corresponding temperature uncertainty is related to the heat capacity of the sensor~\cite{PhysRevLett.114.220405,Campbell_2018,Liu_2019}. On the other scenario, if the sensor-reservoir interaction is too strong, resulting in the so-called bound state effect~\cite{Xiong2015,PhysRevA.89.012128,PhysRevE.90.022122}, or the encoding time is short, the sensor cannot be completely thermalized, which implies the state of the sensor is unstable in the time domain. In this situation, to obtain the information of temperature, one needs to monitor the time evolution of the sensor's reduced density matrix. In comparison with the fully thermalized thermometer, the partly thermalized thermometer may have certain advantages, because the quantum coherence can be utilized as the quantum resource to boost the sensing performance, which means the maximum precision is generally achieved out of the thermal equilibrium regime~\cite{PhysRevA.101.032112,Tamascelli_2020,PhysRevA.103.012217}.

However, many previous studies of the partly thermalized thermometer restricted their attentions to the Born-Markov approximation~\cite{PhysRevA.98.050101,PhysRevA.99.062114,L_pez_V_zquez_2020}, the weak-sensor-reservoir-coupling regime~\cite{PhysRevLett.120.140501}, or the pure dephasing encoding mechanism~\cite{PhysRevResearch.3.013244,PhysRevA.101.032112,PhysRevA.103.012217}. Such treatments can provide an analytical result as well as a intuitionistic picture, but inevitably ignore certain important physical phenomena. For example, as reported in Refs.~\cite{PhysRevA.88.035806,PhysRevA.102.032607,PhysRevA.103.L010601,PhysRevApplied.15.054042}, the authors demonstrated that the non-Markovian effect can severely influence the performance of quantum sensing; and the authors of Refs.~\cite{Tamascelli_2020,PhysRevA.102.032607,SALARISEHDARAN2019126006} reported that the pure dephasing mechanism is not the optimal sensor-reservoir encoding scenario. Thus, to obtain a global view and more physical insights into the quantum thermometry, going beyond the above traditional paradigms is highly desirable.

To address these concerns, we employe the hierarchical equations of motion (HEOM) approach~\cite{doi:10.1143/JPSJ.58.101,PhysRevA.41.6676,doi:10.1063/5.0011599,PhysRevE.75.031107,doi:10.1063/1.2713104,doi:10.1063/1.2938087,PhysRevA.98.012110,PhysRevA.98.032116}, which is a non-perturbatively numerical method, to investigate the effect of the reservoir's characteristic, as well as the form of the sensor-reservoir coupling operator on the performance of a single-qubit quantum thermometer in both partly and fully thermalized situations.

This paper is organized as follows. In Sec.~\ref{sec:sec2}, we briefly outline some basic formalism about the quantum parameter estimation. In Sec.~\ref{sec:sec4}, we employe the HEOM method to investigate the sensitivity of a single-qubit quantum thermometry. The conclusion of this paper is drawn in Sec.~\ref{sec:sec5}. In the two appendixes, we provide some additional materials about the HEOM method, the Born-Markov master equation approach, as well as the variational polaron transformation. Throughout the paper, we set $\hbar=k_{\mathrm{B}}=1$ for the sake of simplicity.

\section{Quantum parameter estimation}\label{sec:sec2}

In this section, we would like to recall some basic concepts as well as some important formalism in quantum parameter estimation theory. Generally speaking, to sense a physical quantity $\lambda$ labeling on a quantum system, one first needs a quantum sensor, which is initially prepared in a suitable input state $\varrho_{\mathrm{in}}$, and coupled to the sensor to the system of interest. Due to the sensor-system interaction, the message about $\lambda$ can be encoded into the output state of the sensor via a $\lambda$-dependent mapping $\varrho_{\mathrm{out}}=\mathcal{M}_{\lambda}(\varrho_{\mathrm{in}})=\varrho_{\lambda}$. Such an encoding process can be realized by a unitary rotation~\cite{Hauke2016,PhysRevB.99.045117,PhysRevLett.124.060402} or a nonunitary reduced dynamics~\cite{PhysRevA.102.032607,PhysRevA.103.L010601,PhysRevApplied.15.054042}.

As long as the output state $\varrho_{\lambda}$ is obtained, the information about $\lambda$ can be extracted by measuring the sensor's observable. More specifically speaking, one can construct a measurement operator $\mathcal{\hat{O}}$ and calculate the expected value $\mathcal{\bar{O}}\equiv \mathrm{Tr}(\varrho_{\lambda}\mathcal{\hat{O}})$ as well as the variance $\mathcal{\check{O}}^{2}\equiv \overline{\mathcal{O}^{2}}-\mathcal{\bar{O}}^{2}$. Then, the sensing precision with respect to $\{\varrho_{\lambda},\mathcal{\hat{O}}\}$ can be evaluated via the standard error propagation formula~\cite{PhysRevLett.79.3865,PhysRevLett.123.040402} $\delta^{2}\lambda(\mathcal{\hat{O}})=\mathcal{\check{O}}^{2}/|\partial_{\lambda}\mathcal{\bar{O}}|^{2}$. Running over all the possible measurement schemes, one can find the optimal measurement operator $\mathcal{\hat{O}}_{\mathrm{opt}}$ corresponding to the ultimate sensing precision. The ultimate sensing precision is constrained by the famous quantum Cram$\mathrm{\acute{e}}$r-Rao bound~\cite{PhysRevLett.72.3439,Liu_2019}
\begin{equation}\label{eq:eq1}
\delta^{2}\lambda\geq \delta^{2}\lambda(\mathcal{\hat{O}}_{\mathrm{opt}})=\mathcal{F}_{\lambda}^{-1},
\end{equation}
where $\mathcal{F}_{\lambda}\equiv \mathrm{Tr}(\hat{\varsigma}^{2}\varrho_{\lambda})$ with $\hat{\varsigma}$ determined by $\partial_{\lambda}\varrho_{\lambda}=\frac{1}{2}(\hat{\varsigma}\varrho_{\lambda}+\varrho_{\lambda}\hat{\varsigma})$ is the quantum Fisher information (QFI)~\cite{Liu_2019}. Specifically, if the output state $\varrho_{\lambda}$ is a two-dimensional density matrix described in the Bloch representation, namely, $\varrho_{\lambda}=\frac{1}{2}(\mathbf{1}_{2}+\pmb{r}\cdot\pmb{\hat{\sigma}})$ with $\pmb{r}$ being the Bloch vector and $\pmb{\hat{\sigma}}\equiv(\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z})$ being the vector of Pauli matrices, the QFI can be easily calculated via~\cite{PhysRevA.87.022337,Liu_2019}
\begin{equation}\label{eq:eq2}
\mathcal{F}_{\lambda}=|\partial_{\lambda}\pmb{r}|^{2}+\frac{(\pmb{r}\cdot\partial_{\lambda}\pmb{r})^{2}}{1-|\pmb{r}|^{2}}.
\end{equation}
When $\varrho_{\lambda}$ is a pure state, Eq.~(\ref{eq:eq2}) further reduces to $\mathcal{F}_{\lambda}=|\partial_{\lambda}\pmb{r}|^{2}$. Physically speaking, the error propagation formula tells us how to extract the message about $\lambda$ from the output state $\varrho_{\lambda}$ with respect to the given measurement operator $\mathcal{\hat{O}}$, while $\mathcal{F}_{\lambda}$ fully quantifies the most statistical information about $\lambda$ contained in the output state $\varrho_{\lambda}$. There is no general way to find the optimal measurement scheme saturating the best attainable precision determined by the QFI. In this sense, how to construct the optimal measurement scheme is of importance in the research of quantum sensing.

\section{Our scheme}\label{sec:sec4}

\begin{figure}
\centering
\includegraphics[angle=0,width=8cm]{Figure1}
\caption{Schematic diagram of our quantum thermometry scheme in which a two-level system is used to as the sensor estimate the temperature of a bosonic reservoir.}\label{fig:fig1}
\end{figure}

In this section, we propose our scheme and analyze its sensing efficiency. A two-level system or a qubit, acting as the quantum sensor, is employed to estimate the temperature of a bosonic reservoir (see Fig.~\ref{fig:fig1}). The total Hamiltonian of the sensor plus the finite-temperature reservoir is described as follows:
\begin{equation}\label{eq:eq4}
\hat{H}=\hat{H}_{\mathrm{s}}+\sum_{k}\omega_{k}\hat{b}_{k}^{\dagger}\hat{b}_{k}+\mathcal{\hat{S}}\otimes\sum_{k}g_{k}(\hat{b}_{k}^{\dagger}+\hat{b}_{k}),
\end{equation}
where $\hat{H}_{\mathrm{s}}=\frac{1}{2}\epsilon\hat{\sigma}_{z}+\frac{1}{2}\Delta\hat{\sigma}_{x}$ is the Hamiltonian of the sensor with $\epsilon$ being the bias and $\Delta$ being the tunneling parameter, operators $\hat{b}_{k}$ and $\hat{b}_{k}^{\dagger}$ are annihilation and creation operators of the $k$th bosonic mode with frequency $\omega_{k}$, respectively. The parameter $g_{k}$ labels the coupling strength between the sensor and the $k$th bosonic mode. Generally, it is very convenient to encode the frequency dependence of the interaction strengths in the so-called spectral density, which is defined by $J(\omega)\equiv\sum_{k}g_{k}^{2}\delta(\omega-\omega_{k})$. In this paper, we consider an Ohmic spectral density with a Drude-type cutoff
\begin{equation}\label{eq:eq5}
J(\omega)=\frac{2}{\pi}\frac{\chi\omega\omega_{\mathrm{c}}}{\omega^{2}+\omega_{\mathrm{c}}^{2}},
\end{equation}
where $\chi$ qualifies the sensor-reservoir coupling strength, and $\omega_{\mathrm{c}}$ denotes the cutoff frequency.

In our scheme, we consider a general coupling operator, i.e., $\mathcal{\hat{S}}$ has the form of~\cite{doi:10.1021/acs.jpclett.0c00985}
\begin{equation}\label{eq:eq6}
\mathcal{\hat{S}}=\hat{\sigma}_{\theta}\equiv\sin\theta\hat{\sigma}_{z}+\cos\theta\hat{\sigma}_{x}.
\end{equation}
By varying the coupling angle $\theta$, all the directions on the $x$-$z$ plane of the Bloch sphere can be taken into account. In Refs.~\cite{doi:10.1063/1.4950888,doi:10.1063/1.4825205,doi:10.1063/5.0027976,doi:10.1080/00268976.2018.1430385}, $\sin\theta\hat{\sigma}_{z}$ is named as the diagonal coupling term, while $\cos\theta\hat{\sigma}_{x}$ is called the off-diagonal coupling operator. However, such a naming convention may lead to misunderstanding when the diagonal and the off-diagonal terms are included in both $\hat{H}_{\mathrm{s}}$ and $\mathcal{\hat{S}}$ at the same time. To avoid such a problem, in this paper, we prefer to use the $z$-type and $x$-type coupling terms to indicate $\sin\theta\hat{\sigma}_{z}$ and $\cos\theta\hat{\sigma}_{x}$ in $\mathcal{\hat{S}}$, respectively. Due to the rotational symmetry of $\hat{\sigma}_{\theta}$, we restrict our study to $0\leq\theta<\pi$. If $\theta=\pi/2$, a more $z$-type coupling operator is included in $\mathcal{\hat{S}}$; when $\theta=0$, only the $x$-type coupling is considered~\cite{doi:10.1063/1.4950888}. As demonstrated in Refs.~\cite{doi:10.1063/1.4950888,doi:10.1063/1.3697817,doi:10.1063/1.4825205}, the authors demonstrated the $x$-type coupling term plays a positive role in restraining coherence and can lead to a much richer ground-state phase diagram in the spin-boson systems. These results inspire us to explore the effect of $z$-type and $x$-type coupling on sensitivity of quantum thermometry.

To realize our sensing scheme, we assume the whole sensor-reservoir system is initially prepared in a product state
\begin{equation}\label{eq:eq7}
\begin{split}
\varrho_{\mathrm{sb}}(0)=&\varrho_{\mathrm{s}}(0)\otimes\varrho_{\mathrm{b}}^{\mathrm{G}}\\
=&|\psi_{\mathrm{s}}(0)\rangle\langle\psi_{\mathrm{s}}(0)|\otimes\frac{\exp(-\beta\hat{H}_{\mathrm{b}})}{\mathrm{Tr}_{\mathrm{b}}[\exp(-\beta\hat{H}_{\mathrm{b}})]},
\end{split}
\end{equation}
where $|\psi_{\mathrm{s}}(0)\rangle=\cos\alpha|\mathrm{e}\rangle+\sin\alpha e^{-i\varphi}|g\rangle$ with $|\mathrm{e},g\rangle$ being the eigenstates of the Pauli $z$-operator and $\hat{H}_{\mathrm{b}}=\sum_{k}\omega_{k}\hat{b}_{k}^{\dagger}\hat{b}_{k}$ denotes the Hamiltonian of the reservoir. The parameter $\beta\equiv T^{-1}$ denotes the inverse temperature and is the quantity of interest to be estimated in this paper.

During the purely numerical calculations to the QFI, one needs to handle the first-order derivative to the parameter $\beta$, say $\partial_{\beta}\pmb{r}$ in Eq.~(\ref{eq:eq2}). In this paper, the first-order derivative for an arbitrary $\beta$-dependent function $f_{\beta}$ is numerically done by adopting the following finite difference method:
\begin{equation}\label{eq:eq8}
\partial_{\beta} f_{\beta}\simeq\frac{-f_{\beta+2\delta}+8f_{\beta+\delta}-8f_{\beta-\delta}+f_{\beta-2\delta}}{12\delta}.
\end{equation}
We set $\delta/\beta=10^{-6}$, which provides a very good accuracy.

\subsection{Non-Markovian temperature sensing}\label{sec:sec4a}

In this subsection, we first investigate the dynamical behavior of QFI within an exact non-Markovian framework. Depending on the characteristic of the reservoir spectral density, quantum reservoirs can be roughly classified into two different categories: the fast reservoir $\omega_{\mathrm{c}}>\max\{\epsilon,\Delta\}$ and the slow reservoir $\omega_{\mathrm{c}}\leq\max\{\epsilon,\Delta\}$~\cite{doi:10.1080/00268976.2018.1430385,doi:10.1063/1.4722336,https://doi.org/10.1002/wcms.1375}. An interesting question arises here: what is the influence of the reservoir's characteristic on the sensing performance? In this subsection, we try to address this question.

\begin{figure}
\centering
\includegraphics[angle=0,width=0.35\textwidth]{Figure2}
\caption{The dynamics QFI with different cutoff frequencies: $\omega_{\mathrm{c}}=0.5\Delta$ (blue rectangles), $\omega_{\mathrm{c}}=0.8\Delta$ (red circles), $\omega_{\mathrm{c}}=4\Delta$ (purple rhombuses) and $\omega_{\mathrm{c}}=10\Delta$ (magenta triangles). The inset figure depicts the steady-state QFI in the long-time regime. Parameters are chosen as $\alpha=\pi/4$, $\varphi=\pi/2$, $\epsilon=0.5\Delta$, $\beta\Delta=0.06$, $\theta=0$ and $\chi=0.06\Delta$ with $\Delta=1\,\mathrm{cm}^{-1}$.}\label{fig:fig2}
\end{figure}

We first consider the so-called deep fast-reservoir regime, i.e., $\omega_{\mathrm{c}}\gg\max\{\epsilon,\Delta\}$, in which the reservoir is memoryless (see Fig.~\ref{fig:fig6} in Appendix A and Refs.~\cite{doi:10.1080/00268976.2018.1430385,doi:10.1063/1.4722336,https://doi.org/10.1002/wcms.1375,PhysRevA.86.012115} for more details). In this regime, the dynamical behavior of QFI from the HEOM approach is displayed by the magenta triangles in Fig.~\ref{fig:fig2}. One can see the QFI gradually increases from its initial value $\mathcal{F}_{\beta}(0)=0$ to the maximum value. Then, the QFI begins to decrease and eventually tends to a steady value in the long-encoding-time limit. Such behavior of $\mathcal{F}_{\beta}(t)$ is physically reasonable because the whole temperature-sensing process includes the following three steps. First, no message about the temperature is contained in the initial state of the sensor, resulting in $\mathcal{F}_{\beta}(0)=0$ at the beginning. Then, the sensor-reservoir interaction, which generates the temperature's information in $\varrho_{\mathrm{s}}(t)$ with the evolution of encoding time, leads to the increase of QFI. Finally, in the long-time limit, the value of $\mathcal{F}_{\beta}(t)$ remains unchanged because, in this step, the message about the temperature is completely from the steady-state reduced density matrix $\varrho_{\mathrm{s}}(\infty)$, which is independent of the encoding time. The above result means there exists an optimal encoding time which can maximize the value of QFI. The appearance of maximal QFI can be understood as the physical result of the interplay between the encoding and the decoherence ~\cite{PhysRevA.102.032607,PhysRevA.103.L010601,PhysRevApplied.15.054042,PhysRevA.97.012125}. In Fig.~\ref{fig:fig3}, we plot the maximal QFI, $\max_{t}\mathcal{F}_{\beta}$, as a function of the coupling angle $\theta$. One can see the maximal QFI is quite sensitive to the coupling angle. Using our above results, one can design the most efficient sensor-reservoir interaction Hamiltonian to obtain the maximum precision estimation by adjusting the weight of the $x$-type coupling operator in the whole sensor-reservoir interaction Hamiltonian.

\begin{figure}
\centering
\includegraphics[angle=0,width=0.35\textwidth]{Figure3}
\caption{The maximum QFI with respect to the optimal encoding time versus the coupling angle for different cutoff frequencies: $\omega_{\mathrm{c}}=0.5\Delta$ (blue rectangles), $\omega_{\mathrm{c}}=0.8\Delta$ (red circles), $\omega_{\mathrm{c}}=4\Delta$ (purple rhombuses) and $\omega_{\mathrm{c}}=10\Delta$ (magenta triangles). Other parameters are the same as those of Fig.~\ref{fig:fig2}.}\label{fig:fig3}
\end{figure}

Next, we consider the slow-reservoir regime, for example, the blue rectangles with $\omega_{\mathrm{c}}=0.5\Delta$ in Fig.~\ref{fig:fig2}, in which the reservoir-induced decoherence should be strongly non-Markovian~\cite{doi:10.1080/00268976.2018.1430385,doi:10.1063/1.4722336,https://doi.org/10.1002/wcms.1375}. In this situation, we find the dynamics of QFI presented by the HEOM method exhibits a collapse-and-revival phenomenon resulting in multiple local maximums, which is completely different from that of the fast reservoir case. A similar result was also reported in Refs.~\cite{PhysRevA.88.035806,PhysRevA.102.032607} and can be regarded as evidence of the information's backflow from the reservoir back to the sensor. Moreover, we also display the maximum QFI with respect to the optimal encoding time versus the coupling angle in Fig.~\ref{fig:fig3}. Multiple local-most-efficient coupling angles are revealed (see the the blue rectangles in Fig.~\ref{fig:fig3}). Additionally, we find the maximal QFI in the slow-reservoir regime can be much larger than that of the fast-reservoir case for the entire range of $0\leq\theta<\pi$. This result suggests the non-Markovian effect, generated by the characteristics of slow reservoir, may be used to improve the estimation precision.

\subsection{Breakdown of the canonical statistics}\label{sec:sec4b}

In this subsection, we shall discuss the feature of steady-state QFI in the long-encoding-time regime. If the sensor-reservoir coupling is weak, the long-time steady state of a fully thermalized thermometer can be described by the canonical Gibbs state, namely, $\varrho_{\mathrm{s}}(\infty)\simeq\varrho_{\mathrm{s}}^{\mathrm{G}}=\exp(-\beta \hat{H}_{\mathrm{s}})/\mathrm{Tr}[\exp(-\beta \hat{H}_{\mathrm{s}})]$. In such an approximate treatment, by diagonalizing $\hat{H}_{\mathrm{s}}$, one can easily find the ratio of $\varrho_{\mathrm{gg}}(\infty)/\varrho_{\mathrm{ee}}(\infty)$ is a monotonously and exponentially increasing function
\begin{equation}\label{eq:eq9}
\frac{\varrho_{\mathrm{gg}}(\infty)}{\varrho_{\mathrm{ee}}(\infty)}\simeq\frac{\varrho_{\mathrm{gg}}^{\mathrm{G}}}{\varrho_{\mathrm{ee}}^{\mathrm{G}}}=e^{\beta\Omega},
\end{equation}
where $\Omega=\sqrt{\epsilon^{2}+\Delta^{2}}$ shall be regarded as the sensor's Rabi frequency. And the corresponding QFI with respect to the canonical Gibbs state is given by
\begin{equation}\label{eq:eq10}
\mathcal{F}_{\beta}^{\mathrm{G}}=\frac{\Omega^{2}}{2+2\cosh(\beta\Omega)}.
\end{equation}
From the above expression, one can easily find $\mathcal{F}_{\beta}^{\mathrm{G}}(\Omega)$ is not a monotonic function. With a fixed temperature, $\mathcal{F}_{\beta}^{\mathrm{G}}(\Omega)$ is a increasing function for $\Omega\in[0,\Omega^{*})$ with $\Omega^{*}$ being determined by $\partial_{\Omega}\mathcal{F}_{\beta}^{\mathrm{G}}(\Omega)=0$; while, for $\Omega\in[\Omega^{*},\infty)$, $\mathcal{F}_{\beta}^{\mathrm{G}}(\Omega)$ monotonically decreases. The above analysis means the steady-state behaviors of the population ratio and the QFI strongly depend on the sensor's Rabi frequency $\Omega$.

However, as demonstrated in many previous articles~\cite{doi:10.1063/1.4722336,PhysRevE.80.041106,PhysRevE.84.011114,PhysRevE.84.051112}, in the deep slow-reservoir regime, which generally induces a strongly non-Markovian effect, the reservoir-induced decoherence can give rise to a drastic frequency renormalization, namely, $\Omega\rightarrow\tilde{\Omega}$. The renormalized frequency $\tilde{\Omega}$ is smaller than the original frequency $\Omega$ and their deviation becomes larger as the increase of the sensor-reservoir coupling~\cite{doi:10.1063/1.4722336,PhysRevE.80.041106,PhysRevE.84.011114,PhysRevE.84.051112,doi:10.1063/1.447055}. This result can be qualitatively understood by performing a polaron transformation. Making use of the polaron transformation, as displayed in Appendix B, the sensor's Rabi frequency is renormalized as
\begin{equation}\label{eq:eq12}
\tilde{\Omega}_{\mathrm{P}}=\sqrt{\tilde{\epsilon}^{2}+\eta^{2}\tilde{\Delta}^{2}},
\end{equation}
where $\tilde{\epsilon}=\epsilon\sin\theta+\Delta\cos\theta$, $\tilde{\Delta}=\Delta\sin\theta-\epsilon\cos\theta$, and $\eta$ is the renormalized factor, whose explicit expression is given by Eq.~(\ref{eq:eqb8}), induced by the polaron transformation. In Table~\ref{table:table1}, we display the frequency renormalization within the framework of polaron transformation theory $\tilde{\Omega}_{\mathrm{P}}/\Omega$ versus the sensor-reservoir coupling strength $\chi$. It is clear to see a larger $\chi$ leads to a smaller $\tilde{\Omega}_{\mathrm{P}}/\Omega$. Such a frequency renormalization phenomenon is responsible for the so-called localized-delocalized transition in the well-known spin-boson model~\cite{PhysRevE.80.041106,PhysRevE.84.011114}.

\begin{table}[t]
\begin{center}
\caption{The frequency renormalization $\tilde{\Omega}_{\mathrm{P}}$ and $\tilde{\Omega}_{\mathrm{H}}$ versus the sensor-reservoir coupling strength $\chi$. Parameters are chosen as $\alpha=\pi/4$, $\varphi=\pi/2$, $\theta=2\pi/3$, $\epsilon=2\Delta$, $\beta\Delta=0.95$, $\omega_{\mathrm{c}}=0.8\Delta$, and $\Delta=0.1\mathrm{cm}^{-1}$.}\label{table:table1}
\setlength{\tabcolsep}{7.25pt}
\begin{tabular}{ccccccc}
\hline
  \hline
  $\chi/\Delta$ & 0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5\\
  $\tilde{\Omega}_{\mathrm{P}}/\Omega$ & 1.000 & 0.968 & 0.932 & 0.890 & 0.837 & 0.741\\
  $\tilde{\Omega}_{\mathrm{H}}/\Omega$ & 1.000 & 0.834 & 0.818 & 0.794 & 0.766 & 0.734\\
  \hline
  \hline
\end{tabular}
\end{center}
\end{table}

\begin{figure}
\centering
\includegraphics[angle=0,width=0.49\textwidth]{Figure4}
\caption{Schematic diagram of the influence of frequency renormalization $\Omega\rightarrow\tilde{\Omega}$ on (a) the steady-state population ratio and (b) the QFI. (c) The steady-state population ratio $\varrho_{\mathrm{gg}}(\infty)/\varrho_{\mathrm{ee}}(\infty)$ is plotted as a function of the sensor-reservoir coupling strength $\chi$. (d) The steady-state QFI versus $\chi$. The red five-point stars are the numerical results obtained by the HEOM method, the blue rectangles are the steady-state solutions from the Born-Markov master equation approach, while the blue solid lines represent the results from the canonical Gibbs state. Parameters are chosen as $\alpha=\pi/4$, $\varphi=\pi/2$, $\epsilon=0.5\Delta$, $\beta\Delta=5$, and $\omega_{\mathrm{c}}=0.5\Delta$ with $\Delta=50\,\mathrm{cm}^{-1}$. Here, $\Omega^{*}\simeq24\,\mathrm{cm}^{-1}$ and $\Omega\simeq 56\,\mathrm{cm}^{-1}$.}\label{fig:fig4}
\end{figure}

Thus, the most simple and straightforward correction to the previous assumption of canonical statistics is replacing the original Rabi frequency $\Omega$ by a smaller renormalized frequency $\tilde{\Omega}$. Under such treatment, one can see the frequency shift $\Omega\rightarrow\tilde{\Omega}$ shall lead to a smaller value of $\varrho_{\mathrm{gg}}(\infty)/\varrho_{\mathrm{ee}}(\infty)$ [see Figs.~\ref{fig:fig4}(a) and 5(a)] compared with that of the canonical Gibbs state case, which implies the breakdown of canonical statistics~\cite{Zhu_2020,PhysRevA.90.032114,doi:10.1063/1.4940218}. Moreover, if $\Omega>\Omega^{*}$, the frequency renormalization results in a larger QFI in comparison with that of the canonical statistics [see Fig.~\ref{fig:fig4}(b)]. On the contrary, if $\Omega<\Omega^{*}$, the frequency renormalization in turn diminishes the sensing precision [see Fig.~\ref{fig:fig5}(b)].

\begin{figure}
\centering
\includegraphics[angle=0,width=0.49\textwidth]{Figure5}
\caption{The same with Fig.~\ref{fig:fig4}, but $\beta\Delta=1$ and $\Delta=10\,\mathrm{cm}^{-1}$. Here, $\Omega^{*}\simeq24\,\mathrm{cm}^{-1}$ and $\Omega\simeq 11\,\mathrm{cm}^{-1}$.}\label{fig:fig5}
\end{figure}

The above analytical analysis is completely based on the polaron transformation, which only provides a qualitative interpretation as well as a pictorial illustration. To obtain a quantitative result, we display the steady-state population ratio and the QFI as functions of the coupling strength from the HEOM and the Born-Markov master equation methods (see Appendix A for details) in Figs.~\ref{fig:fig4}(c,) and 4(d) and Figs.~\ref{fig:fig5}(c,) and 5(d). As a result of neglecting the higher-order terms of the sensor-reservior interaction, the steady-state solutions from the Born-Markov master equation approach coincide with the canonical statistics, i.e., the steady-state population ratio and the QFI are independent of the coupling strength. In sharp contrast to this, one can see the numerical results obtained by the HEOM method are in good agreement with our previous discussions about the frequency renormalization within the framework of the polaron transformation. In Table~\ref{table:table1}, we also display the renormalized Rabi frequency predicted by the HEOM method, namely $\tilde{\Omega}_{\mathrm{H}}\equiv\beta^{-1}\ln[\varrho_{\mathrm{gg}}(\infty)/\varrho_{\mathrm{ee}}(\infty)]$, as the function of the coupling strength $\chi$. It is clear to see the evolutionary trend of $\tilde{\Omega}_{\mathrm{H}}/\Omega$ in qualitative agreement with that of $\tilde{\Omega}_{\mathrm{P}}/\Omega$. These results demonstrate, in the strong-coupling regime, the noncanonical distribution occurs, which plays a complicated role in the quantum thermometry. The result presented by the red stars in Figs.~\ref{fig:fig4}(c,) and 4(d) and Figs.~\ref{fig:fig5}(c,)and 5(d) cannot be predicted by using the Gorini-Kossakowski-Sudarshan-Lindblad master equation formula or other common perturbative methods, where the effect of frequency shift is generally washed out~\cite{HUANG2010256,PhysRevA.94.062116}.

Therefore, we uncover a threshold $\Omega=\Omega^{*}$, above such critical Rabi frequency, i.e., $\Omega>\Omega^{*}$, the sensing precision can be effectively enhanced by strengthening the sensor-reservoir coupling. In this sense, the sensitivity of quantum thermometry can be improved by engineering the sensor's bare frequency. Here, we concentrate the frequency renormalization mechanism on the slow-reservoir regime, the reason is twofold. First, in the deep fast-reservoir regime, the decoherence is Markovian which means the canonical statistics assumption is valid with no need for corrections. Second, as $\omega_{\mathrm{c}}/\max\{\epsilon,\Delta\}\rightarrow\infty$, the renormalized factor $\eta$ approaches to $1$ accordingly, resulting in the disappearance of frequency shift phenomenon in the deep-reservoir regime. Our result is consistent with that of Ref.~\cite{PhysRevA.96.062103}, in which the authors found the sensing precision of a Brownian thermometer can be improved by increasing the coupling at low temperature. Their result can be easily understood by using our threshold mechanism: the critical Rabi frequency $\Omega^{*}$ approaches to zero in the low-temperature limit, which means the condition $\Omega>\Omega^{*}$ becomes very easy to be satisfied. Thus, as long as the Brownian sensor has a finite bare frequency, i.e., $\Omega>0$, the sensor-reservoir can boost the thermometric precision without doubt. Of course, it is worth mentioning that our scenario is totally different from that found in Ref.~\cite{PhysRevA.96.062103}. Their quantum thermometry is based on the Caldeira-Leggett Hamiltonian, namely, the sensor in Ref.~\cite{PhysRevA.96.062103} is a continuous-variable system, rather than the discrete-variable sensor (qubit) in this paper.

\section{Conclusion}\label{sec:sec5}

In summary, going beyond the usual assumptions of the Born-Markov theory, the pure dephasing mechanism and the weak-coupling approximation, we propose a single-qubit thermometer scheme and analyze its sensing performance with the help of the HEOM approach. We find, when the encoding time is short and the sensor is partly thermalized, the non-Markovian effect induced by the slow reservoir's characteristics may enhance the efficiency of the quantum thermometry. By optimizing the encoding time, we investigate the relation between the maximal QFI and the sensor-reservoir coupling angle. It is revealed that the performance of our quantum thermometry can be further boosted by modulating the weight of the $x$-type coupling operator in the whole sensor-reservoir Hamiltonian. Moreover, in the slow-reservoir regime, by strengthening the sensor-reservoir coupling, we find the noncanonical feature appears in the sensor's steady state and the corresponding QFI obtained by the HEOM method can be larger than the value predicated by the canonical Gibbs state as well as the Born-Markov master equation approach, under suitable conditions. A possible threshold mechanism, which is based on the frequency renormalization appearing in the strongly non-Markovian regime, is proposed to explain the above result in the long-encoding-time limit. Finally, due to the generality of the qubit-based quantum thermometer, we expect our results to be of interest for certain potential applications in the researches of quantum metrology and quantum sensing.

\section{Acknowledgments}

The authors thank Dr. Si-Yuan Bai, Professor Jun-Hong An and Professor Hong-Gang Luo for many fruitful discussions. W. Wu wishes to especially thank Dr. Chong Chen and an anonymous referee for their suggestions on the discussion of frequency renormalization. The work was supported by the National Natural Science Foundation (Grant No. 11704025).

\section{Appendix A: HEOM}

The HEOM method is a purely numerical technique, which exactly maps the Schr$\ddot{\mathrm{o}}$dinger equation or quantum Liouville equation to a set of ordinary differential equations by introducing auxiliary density matrices. These ordinary differential equations can be treated numerically by the Runge-Kutta algorithm. Without invoking the Born-Markov, weak-coupling and rotating-wave approximations, the HEOM can provide a rigorous numerical dynamics for an open quantum system~\cite{doi:10.1143/JPSJ.58.101,PhysRevA.41.6676,doi:10.1063/5.0011599,PhysRevE.75.031107,doi:10.1063/1.2713104,doi:10.1063/1.2938087,PhysRevA.98.012110,PhysRevA.98.032116}.

To realize the above HEOM method, it is generally required that the reservoir correlation function $C(t)$, which is defined by
\setcounter{equation}{0}
\renewcommand\theequation{A\arabic{equation}}
\begin{equation}\label{eq:eqa1}
\begin{split}
C(t)=&\mathrm{Tr}_{\mathrm{b}}\Big{(}e^{it\hat{H}_{\mathrm{b}}}\mathcal{\hat{B}}e^{-it\hat{H}_{\mathrm{b}}}\mathcal{\hat{B}}\Big{)}\\
=&\int_{0}^{\infty} d\omega J(\omega)\Big{[}\coth\Big{(}\frac{\beta\omega}{2}\Big{)}\cos(\omega t)-i\sin(\omega t)\Big{]},
\end{split}
\end{equation}
can be expressed as a sum of exponential functions~\cite{doi:10.1143/JPSJ.58.101,PhysRevA.41.6676,doi:10.1063/5.0011599,PhysRevE.75.031107,doi:10.1063/1.2713104,doi:10.1063/1.2938087,PhysRevA.98.012110,PhysRevA.98.032116,doi:10.1063/1.4893931}. Here, $\mathcal{\hat{B}}\equiv\sum_{k}g_{k}(\hat{b}_{k}^{\dagger}+\hat{b}_{k})$. Fortunately, for the Ohmic spectral density considered in this paper, the reservoir correlation function satisfies such requirement. Substituting Eq.~(\ref{eq:eq5}) into Eq.~(\ref{eq:eqa1}), one can find
\begin{equation}\label{eq:eqa2}
C(t)=\sum_{\ell=0}^{\varepsilon}\zeta_{\ell}e^{-\upsilon_{\ell}t},
\end{equation}
where $\upsilon_{\ell}=\omega_{\mathrm{c}}\delta_{0\ell}+2\ell\pi(1-\delta_{0\ell})\beta^{-1}$ denotes the $\ell$th Matsubara frequency and
\begin{equation}
\begin{split}
\zeta_{\ell}=&\frac{4\chi\omega_{\mathrm{c}}}{\beta}\frac{\upsilon_{\ell}}{\upsilon_{\ell}^{2}-\omega_{\mathrm{c}}^{2}}(1-\delta_{0\ell})\\
&+\Big{[}\chi\omega_{\mathrm{c}}\cot\Big{(}\frac{\beta\omega_{\mathrm{c}}}{2}\Big{)}-i\chi\omega_{\mathrm{c}}\Big{]}\delta_{0\ell},
\end{split}
\end{equation}
are the expansion coefficients. Here, we set a truncation parameter $\varepsilon$ to ensure the above series remain finite. With the help of the Eq.~(\ref{eq:eqa2}), the hierarchical equations can be constructed following detailed exposition in Refs.~\cite{PhysRevA.98.012110,PhysRevA.98.032116}. The HEOM reads
\begin{equation}
\begin{split}
\frac{d}{dt}\varrho_{\vec{\nu}}(t)=&\Big{(}\mathcal{\hat{L}}_{\mathrm{s}}-\vec{\nu}\cdot\vec{\mu}\Big{)}\varrho_{\vec{\nu}}(t)\\
&+\hat{\Phi}\sum_{\ell=0}^{\varepsilon}\hat{\varrho}_{\vec{\nu}+\vec{e}_{\ell}}(t)+\sum_{\ell=0}^{\varepsilon}\nu_{\ell}\hat{\Theta}_{\ell}\hat{\varrho}_{\vec{\nu}-\vec{e}_{\ell}}(t),
\end{split}
\end{equation}
where $\vec{\nu}=(\nu_{0},\nu_{1},\nu_{2},.~.~ .,\nu_{\varepsilon})$ is a $(\varepsilon+1)$-dimensional index, $\vec{e}_{\ell}=(0,0,0,.~.~.1_{\ell},.~.~.,0)$ and $\vec{\mu}=(\upsilon_{0},\upsilon_{1},\upsilon_{2},.~.~.,\upsilon_{\varepsilon})$ are $(\varepsilon+1)$-dimensional vectors. The superoperators $\mathcal{\hat{L}}_{\mathrm{s}}$, $\hat{\Phi}$, and $\hat{\Theta}_{\ell}$ are defined as $\mathcal{\hat{L}}_{\mathrm{s}}\equiv-i\hat{H}_{\mathrm{s}}^{\times}$, $\hat{\Phi}\equiv-i\mathcal{\hat{S}}^{\times}$ and
\begin{equation}
\hat{\Theta}_{\ell}=-i\Big{[}\mathrm{Re}(\zeta_{\ell})\mathcal{\hat{S}}^{\times}+i\mathrm{Im}(\zeta_{\ell})\mathcal{\hat{S}}^{\circ}\Big{]},
\end{equation}
where $\hat{o}_{1}^{\times}\hat{o}_{2}\equiv\hat{o}_{1}\hat{o}_{2}-\hat{o}_{2}\hat{o}_{1}$ and $\hat{o}_{1}^{\circ}\hat{o}_{2}\equiv\hat{o}_{1}\hat{o}_{2}+\hat{o}_{2}\hat{o}_{1}$.

In numerical simulations, the initial states of the auxiliary operators are given by
\begin{equation}
\varrho_{\vec{\nu}=\vec{0}}(0)=\varrho_{\mathrm{s}}(0);~~\varrho_{\vec{\nu}\neq\vec{0}}(0)=\vec{0},
\end{equation}
where $\vec{0}=(0,0,0. . .,0)$ is a $(\varepsilon+1)$-dimensional zero vector. In this paper, we constantly increasing the number of the differential equations as well as the value of $\varepsilon$ until the final result converges.

\begin{figure*}
\centering
\includegraphics[angle=0,width=0.99\textwidth]{Figure6}
\caption{The dynamics of population difference $\langle\hat{\sigma}_{z}(t)\rangle$ with different cutoff frequencies: (a) $\omega_{\mathrm{c}}=30\Delta$ (the deep fast-reservoir regime), (b) $\omega_{\mathrm{c}}=20\Delta$, (c) $\omega_{\mathrm{c}}=10\Delta$, (d) $\omega_{\mathrm{c}}=0.07\Delta$, and (e) $\omega_{\mathrm{c}}=0.05\Delta$ (the deep slow-reservoir regime). The red circles are numerical results predicted by the HEOM method, while the blue solid lines are from the Born-Markovian master equation approach. Other parameters are chosen as $\alpha=\pi/4$, $\varphi=\pi/2$, $\epsilon=0.5\Delta$, $\chi=0.06\Delta$, $\beta\Delta=0.25$, $\theta=3\pi/8$, and $\Delta=1\,\mathrm{cm}^{-1}$.}\label{fig:fig6}
\end{figure*}

As a benchmark, we also employe the most common Born-Markov master equation to compute the reduced dynamics of the sensor. As displayed in Ref.~\cite{PhysRevA.94.062116}, the Born-Markov master equation reads
\begin{equation}\label{eq:eqa6}
\frac{d}{dt}\varrho_{\mathrm{s}}(t)=\Big{(}\mathcal{\hat{L}}_{\mathrm{s}}-\mathcal{\hat{S}}^{\times}\hat{\Upsilon}^{\times}+\mathcal{\hat{S}}^{\times}\hat{\Xi}^{\circ}\Big{)}\varrho_{\mathrm{s}}(t),
\end{equation}
where
\begin{equation*}
\hat{\Upsilon}\equiv\int_{0}^{\infty}dt C_{\mathrm{R}}(t)\mathcal{\hat{S}}(-t),
\end{equation*}
and
\begin{equation*}
\hat{\Xi}\equiv-i\int_{0}^{\infty}dt C_{\mathrm{I}}(t)\mathcal{\hat{S}}(-t).
\end{equation*}
Here, $\mathcal{\hat{S}}(t)\equiv e^{it\hat{H}_{\mathrm{s}}}\hat{\mathcal{S}}e^{-it\hat{H}_{\mathrm{s}}}$ and $C_{\mathrm{R}(\mathrm{I})}(t)$ denotes the real (imaginary) part of the correlation function. By neglecting the imaginary Lamb-shift terms~\cite{PhysRevA.94.062116}, the master equation given by Eq.~(\ref{eq:eqa6}) can be further simplified, which shall be very convenient for numerical simulations in practice.

To verify the feasibility of the HEOM method, we here make a comparison between the purely numerical result obtained by the numerical HEOM method and that of the Born-Markov master equation approach. In Fig.~\ref{fig:fig6}, we display the evolution of population difference
\begin{equation}
\langle\hat{\sigma}_{z}(t)\rangle\equiv\mathrm{Tr}\Big{[}\hat{\sigma}_{z}e^{-i\hat{H}t}\varrho_{\mathrm{s}}(0)\otimes\varrho_{\mathrm{b}}^{\mathrm{G}}e^{i\hat{H}t}\Big{]}.
\end{equation}
Good agreement is found between results from the two different approaches in the deep fast-reservoir regime, see Fig.~\ref{fig:fig6}(a). On the contrary, a relatively large deviation is found in the deep slow-reservoir regime, see Figs.~\ref{fig:fig6}6(d) and 6(e). Moreover, we observe that such a deviation becomes more and more evident as $\omega_{\mathrm{c}}/\Delta$ approaches to zero with fixed $\epsilon=0.5\Delta$. This result is physically reasonable, because the reservoir is memoryless in the deep fast-reservoir regime. While, with the decrease of $\omega_{\mathrm{c}}/\Delta$, the non-Markovianity becomes non-negligible~\cite{PhysRevA.86.012115}, which means the results from the Born-Marovian master equation method are unreliable. Our result is consistent with that of Ref.~\cite{doi:10.1080/00268976.2018.1430385} in which only the $z$-type coupling term is taken into consideration.

\section{Appendix B: Variational polaron transformation}

To obtain the explicit expression of $\tilde{\Omega}_{\mathrm{P}}$, we first need a rotation matrix $\mathcal{\hat{R}}\equiv\exp(-\frac{i}{2}\phi\hat{\sigma}_{y})$ with $\phi=\arctan(\cot\theta)$ to diagonalize $\mathcal{\hat{S}}$. By doing so, the original Hamiltonian $\hat{H}$ can be transformed into $\hat{H}'=\mathcal{\hat{R}}^{\dagger}\hat{H}\mathcal{\hat{R}}$, where
\setcounter{equation}{0}
\renewcommand\theequation{B\arabic{equation}}
\begin{equation}
\hat{H}'=\frac{1}{2}\tilde{\epsilon}\hat{\sigma}_{z}+\frac{1}{2} \tilde{\Delta}\hat{\sigma}_{x}+\hat{H}_{\mathrm{b}}+\hat{\sigma}_{z}\otimes\sum_{k}\tilde{g}_{k}(\hat{b}_{k}^{\dagger}+\hat{b}_{k}),
\end{equation}
with $\tilde{g}_{k}=\sin(\theta+\phi)g_{k}$. For the special scope $0\leq\theta<\pi$ considered in this paper, $\tilde{g}_{k}$ coincidentally reduces to $g_{k}$. Employing such rotation, $\hat{H}'$ now has the standard form of well-known spin-boson model.

Next, we apply the variational polaron transformation to $\hat{H}'$ as $\hat{H}''=\exp(\hat{\mathcal{P}})\hat{H}'\exp(-\hat{\mathcal{P}})$. Here, the polaron generator $\hat{\mathcal{P}}$ is given by~\cite{doi:10.1063/1.4722336}
\begin{equation}
\hat{\mathcal{P}}=\hat{\sigma}_{z}\sum_{k}\frac{\xi_{k}}{\omega_{k}}(\hat{b}_{k}^{\dagger}-\hat{b}_{k}),
\end{equation}
where $\xi_{k}=g_{k}\mathfrak{S}(\omega_{k})$ with
\begin{equation}
\mathfrak{S}(\omega_k)=\bigg{[}1+\frac{\eta^{2}\tilde{\Delta}^{2}}{\omega_{k}\tilde{\Omega}_{\mathrm{P}}}\coth\Big{(}\frac{1}{2}\beta\omega_{k}\Big{)}\tanh\Big{(}\frac{1}{2}\beta\tilde{\Omega}_{\mathrm{P}}\Big{)}\bigg{]}^{-1},
\end{equation}
being self-consistently determined by minimizing the Bogoliubov upper bound for the free energy~\cite{doi:10.1063/1.4722336}. The expression of $\hat{H}''$ can be written as $\hat{H}''=\hat{H}_{\mathrm{s}}''+\hat{H}_{\mathrm{b}}+\hat{H}_{\mathrm{sb}}''$, where $\hat{H}_{\mathrm{s}}''$ denotes the transformed Hamiltonian of the sensor
\begin{equation}
\hat{H}_{\mathrm{s}}''=\frac{1}{2}\tilde{\epsilon}\hat{\sigma}_{z}+\frac{1}{2} \eta\tilde{\Delta}\hat{\sigma}_{x}+\sum_{k}\frac{\xi_{k}}{\omega_{k}}\big{(}\xi_{k}-2g_{k}\big{)}.
\end{equation}
The last term in the above expression is an energy shift induced by the polaron transformation, which has no influence on the sensing accuracy in the assumption of canonical statistics. From the expression of $\hat{H}_{\mathrm{s}}''$, one can immediately find the Rabi frequency is $\tilde{\Omega}_{\mathrm{P}}^{2}=\tilde{\epsilon}^{2}+\eta^{2}\tilde{\Delta}^{2}$, which recovers Eq.~(\ref{eq:eq12}) in the main text. The transformed sensor-reservoir interaction Hamiltonian $\hat{H}_{\mathrm{sb}}''$ can be written as $\hat{H}_{\mathrm{sb}}''=\hat{\sigma}_{x}\mathcal{\hat{B}}_{x}+\hat{\sigma}_{y}\mathcal{\hat{B}}_{y}+\hat{\sigma}_{z}\mathcal{\hat{B}}_{z}$, where
\begin{equation}
\mathcal{\hat{B}}_{x}=\frac{1}{2}\tilde{\Delta}\big{(}\cosh\hat{\Sigma}-\eta\big{)},~~\mathcal{\hat{B}}_{y}=\frac{i}{2}\tilde{\Delta}\sinh\hat{\Sigma},
\end{equation}
\begin{equation}
\begin{split}
\mathcal{\hat{B}}_{z}=\sum_{k}\big{(}g_{k}-\xi_{k}\big{)}\Big{(}\hat{b}_{k}^{\dagger}+\hat{b}_{k}\Big{)},
\end{split}
\end{equation}
with
\begin{equation}
\hat{\Sigma}\equiv\sum_{k}\frac{2\xi_{k}}{\omega_{k}}\Big{(}\hat{b}_{k}^{\dagger}-\hat{b}_{k}\Big{)},
\end{equation}
and
\begin{equation}\label{eq:eqb8}
\begin{split}
\eta\equiv&\mathrm{Tr}\Big{(}\varrho_{\mathrm{b}}^{\mathrm{G}}\cosh\hat{\Sigma}\Big{)}\\
=&\exp\bigg{[}-2\int_{0}^{\infty}d\omega\frac{J(\omega)}{\omega^{2}}\mathfrak{S}(\omega)^{2}\coth\Big{(}\frac{1}{2}\beta\omega\Big{)}\bigg{]}.
\end{split}
\end{equation}

redpajama_set_name: RedPajamaArXiv


text:
Green Energy Chronicles
What you should know about alternative energy.
Navy Engineers Install Solar Panels for West African Community
March 8, 2017 // by Timothy E.
(www.defense.gov)
In the rural West African city of Mbour, Senegal, more than 200 children go to school every day in a community with no electricity.
Naval Sea Systems Command engineers partnered with The Lamps, a nonprofit organization, as part of their "Let There Be Light" campaign, which aims to bring electrical power to regions of the world with no affordable supply.
Peter C. Cho, an electrical engineer with Naval Surface Warfare Center, Carderock Division's marine and aviation division; Bryant H. Kim, from NSWC, Indian Head Division; and Sungshin Kwak, from the Naval Research Laboratory dedicated their personal leave time Feb. 10-19 to visit the city, located about two hours from the capital of Dakar. They installed six solar panels, bringing a source of light to a community that had no means of electricity prior to their visit.
Mbour is just one of the many communities Cho and his fellow engineers have visited over the years. They install what Cho describes as a simple, solar electric power systems consisting of panels, batteries, an inverter and circuit breakers, which brings the community enough power to run indoor lights and small appliances. However, Cho noted, the communities they visit do not have the luxury of appliances of any kind.
Immense Need
"The need in that community, as well as all of the communities we do work in, is immense," Cho said. "Every trip I make, when I come back and arrive at the Dulles International Airport [in Washington], I marvel at what a blessed country the United States truly is.
"One thing that always amazes me is everywhere we go the children always seem to be very happy, even in their situation," he continued. "In Mbour, the children go to school but have no shoes, no sandals, and in the summertime it gets so hot — upwards of 120 degrees — they cannot wear clothes. Their happiness is not measurable with materials."
Cho, who emigrated from South Korea, said one of the main reasons he makes these trips every year is because he was lucky enough to immigrate to the United States and receive a good education.
"The U.S. government educated me from bachelor's [degree] all the way to Ph.D," he said. "I began my career at the Naval Undersea Warfare Center in Newport, Rhode Island. I got together with some of my fellow engineers and figured out how we can pass on this great education and engineering experience to the next generation.
"One way we do this is by providing our skills and experience to the areas where there's no electricity available," he said. "If we can give students even something as simple as light in their schoolhouse and it can improve their chances of succeeding, improve their chances of a better education, then I am happy. That is our motivation."
Team Has Grown
Cho said he worked with a team of four engineers when he started bringing solar energy to communities without electricity. Today, that team has grown to 13. The engineers not only travel on their personal leave time, but often use their own money to buy the solar panels. In some cases, he added, they have worked to subsidize funds from the U.S. Agency for International Development and SEED International, a Christian ministry out of Virginia.
"The biggest benefit to installing solar panels is they do not require a lot of maintenance," Cho said. "A lot of the communities we visit are close to the equator; sun is abundant. On average, two hours of sun will provide around eight hours of electricity."
Cho and his team also have installed solar panels in communities in Peru and Cambodia. Wherever they go, he said, being able to provide something as simple as indoor lighting, they leave knowing their work has improved the quality of life for the people of that community.
Category: Solar
Previous Post: « Pollution-fighting Vertical Forest buildings coming to China
Next Post: Jailhouse shock: Taiwan prison aims to jump-start island's solar power dream »
Middle-Class Families Are Set To Receive Solar Panels With No Upfront Costs In The U.S.
April 28, 2019 By // by Timothy E.
Nearly 300 low-income D.C. residents to get solar panels on their homes
July 28, 2017 By // by Timothy E.
Asian Development Bank Invests $57.7 Million To Develop Renewable Energy In Southeast Asia
Richard Branson: Give Coal Miners Clean Energy Jobs
Chinese province runs on 100% renewable energy for a week to test whether grid can cope

redpajama_set_name: RedPajamaCommonCrawl


Mercury exists in various natural forms globally, and people have been familiar with it throughout the history of human development. Mercury was used as a pigment and a miracle drug for immortality in old times and in clinical thermometers and fluorescent lamps until quite recently. It is a unique metal that has been used in various ways by exploiting its distinctive properties.
Human beings have made effective use of mercury throughout history but they have been exposed to the threat posed by it due to its poisonous properties when misused. Minamata disease in Japan resulted in a tremendous number of victims and health damage in the 1960s, and there are still a number of patients suffering its aftereffects.
Since first being established in 1978, Nippon Instruments Corporation has specialized in analyzing mercury. Efforts to control mercury and prevent the spread of any damage to human health resulting from mercury have just commenced on a global basis but mercury will be present. This has led to an increasing awareness of the significance of accurately measuring and controlling mercury.
We, Nippon Instruments Corporation, are committed to making a contribution to customers through mercury measurements and under the corporate philosophy of "Contribute to the Development of Human Society through Advances Made in Scientific Technology".


text:
Centruroides elegans est une espèce de scorpions de la famille des Buthidae.

Distribution
Cette espèce est endémique du Mexique. Elle se rencontre au Sinaloa, au Nayarit, au Jalisco, au Colima, au Michoacán, au Guerrero, en Oaxaca, au Morelos et dans l'État de Mexico.

Description
L'holotype mesure .

Liste des sous-espèces
Selon The Scorpion Files (11/10/2020) :
 Centruroides elegans edentulus Werner, 1939
 Centruroides elegans elegans (Thorell, 1876)
 Centruroides elegans insularis Pocock, 1902 du Nayarit

Publications originales
 Thorell, 1876 : « Études Scorpiologiques. » Atti della Societa Italiana di Scienze Naturali, ,  (texte intégral).
 Pocock, 1902 : « Arachnida. Scorpiones, Pedipalpi, and Solifugae. » Biologia Centrali-Americana, ,  (texte intégral).
 Werner, 1939 : « Neu Eingange von Skorpionen im Zoologischen Museum in Hamburg. II. Teil. » Festschrift zum 60 Geburtstage von Prof. Dr. Embrik Strand, , .

Liens externes
 The Scorpion Files : Jan Ove Rein, Trondheim, Norwegian University of Science and Technology

Notes et références

Buthidae
Espèce de scorpions (nom scientifique)
Faune endémique du Mexique

redpajama_set_name: RedPajamaWikipedia


Strictly limited edition of 1.000 copies, numbered & signed by Maestro Edward Artemiev.
?A Wonderful Boy or Gvozdik?s Adventure? (1970) - directed by Alexander Orlov & Leonid Pekur, ?Birds Over The City? (1974) - directed by Sergey Nikonenko, ?Lost & Found or Everyone Dreams of A Dog? (1976) - directed by Igor Yasulovich, ?Hello, River! Or I, You, We? (1978) - directed by Igor Yasulovich, Pavel Arsenov & Jury Grigoryev.
?The Forest You Never Will Enter? (1978) - directed by Boris Konunov, ?A Strip Of Uncut Wild Flowers? (1979) - directed by Jury Ilyenko, ?The Detective? (1979) - directed by Vladimir Fokin, ?The Night Of Birth? (1980) - cartoon, directed by Rozaliya Zelma, ?The White Shaman? (1982) - directed by Anatoly Nitochkin.
?Every Hunter Wants To Know? (1985) - directed by Michael Ilyenko, ?A Hundred Joys Or The Book Of Great Discoveries? (1981) - directed by Yaroslav Lupiy, ?Farewell To The Greens Of Summer? (1985) - directed by Elyor Ishmukhamedov, ?The Sun? (1981) - cartoon, directed by Vladimir Samsonov, ?A Girl & A Dolphin? (1979) - cartoon, directed by Rozaliya Zelma, ?A Beauty in Mourning? (1981) - directed by Ermek Shinarbaev, ?A Wonderful Boy or Gvozdik?s Adventure? (1970) - directed by Alexander Orlov & Leonid Pekur - Part II, ?Strawberry? (1983) - directed by Vladimir Samsonov, ?In Private? (1983) - directed by Nikita Mikhalkov, ?Arena? (1967) - directed by Samson Samsonov.
?A Strange Story of Dr. Jekyll & Mr. Hyde? (1985) - directed by Alexander Orlov; ?Inspector Gull? (1979) - directed by Alexander Proshkin; ?Seven Elements? (1984) - directed by Gennady Ivanov; ?Returning From Orbit? (1983) - directed by Alexander Surin; ?The Middle Of Life? (1976) - directed by Olgerd Vorontsov; ?The Complicity In The Murder? (1985) - directed by Valery Uskov, Vladimir Krsnopolskiy; ?Risk Strategy? (1978) - directed by Alexander Proshkin; ?The Inner Circle? (1981) - directed by Andrey Konchalovsky) - 30-31.


The Christian Camargo Blog: Please join @therealcamargo Jan 27th in NYC for Russell Means Honoring!
Please join @therealcamargo Jan 27th in NYC for Russell Means Honoring!
An Honoring of American Indian Activist Russell Means. An evening of world music, readings and special preview of "Conspiracy To Be Free."
All proceeds to benefit the Russell Means T.R.E.A.T.Y. Educational Endowment.


text:
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Feeling oddly balanced today? This may be due to the fact that we are celebrating the unofficial autumnal equinox of 2010 (the day in which the Earth experiences equal periods of darkness and daylight). Keep in mind, however, that this is the UNOFFICIAL autumnal equinox — the day that we accept as the equinox when in reality, it varies annually. Those of you in an industry or design field that depends on natural lighting or solar metrics, this is basically a holiday. For the rest of you, this may become just another beautiful autumn day in the English countryside. Speaking of which, today's feature hails from "God's Own Country" (Yorkshire, England). Known for its green pastures and virgin countryside, Yorkshire is also known for its burgeoning art scene due in part to the artistic styles of one Mr. Gauky.
While there is very little known about the true identity of Mr. Gauky, he describes himself as an Yorkshire-born orphan adopted by a rough-and-tumble gang called "the Slouches" — a group he claims as inspiration. While we at Ballista Magazine are cautiously skeptical of this upbringing (a slight exaggeration, to say the least), we do know that he is a very skilled technician in the fields of illustration, painting, product design, and street art. His work may be a bit morbid in concept, but his work is incredibly detailed and skillfully rendered. Mr. Gauky breathes life into his "misfit creatures", animating his characters into the third dimension.
For more Gauking, make sure to check out Mr. Gauky's website, Twitter, Flickr, and/or Facebook pages. That should keep you busy for quite some time — if not, just stop back tomorrow for our next feature. Who knows, maybe it will even be the winner of the Summer Camp Extravaganza Competition. Happy unofficial autumnal equinox!
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Kevin Conroy has been lending his voice to the Dark Knight for 25 years now, appearing in everything from Batman: The Animated Series to the acclaimed Arkham video games and a host of animated features, including last year's Batman: The Killing Joke and this month's Batman and Harley Quinn, which reunites him with Batman: TAS co-creator Bruce Timm.
Flickering Myth's Amon Warmann was lucky enough to speak to Kevin this past week to discuss Batman and Harley Quinn, during which he revealed that there are still several iconic Batman stories he'd like to tick off his bucket list.
Be sure to check back later today for our full interview with Kevin Conroy and let us know which Batman stories you'd like to see adapted as animated movies.


text:
ife for Lancelot was coming apart with such thoroughness that he wondered why his bicycle was still working. Every day, with never a puncture of a broken spoke, it took him to the office so that he could hear more bad news. Ian Cuthbert was recuperating in a clinic, where, after they had straightened out his spine, it had been found advisable to detoxify him. Lancelot had sent Serena along to extract from the convalescent some amplification to the list of writers who could draw and artists who could write. Whether out of misogynist rancour or a deranged mind, Ian had sent her off in search of a document called Mrs Ruisdael's Diary. She had wasted days on the search and was to waste many days more in useless tears after Lancelot told her that she had been led up the garden path. Brian Hutchings continued to be tardy about supplying his manuscript. Lancelot had put Janice on the job of badgering him but Brian simply refused to talk to her, a standpoint he was able to back up with force, because he had acquired a Janice of his own, called Katja. The thought of those two communicating by telephone aroused visions of the Four Horsemen of the Apocalypse riding around your living room. Meanwhile Lancelot's tax case continued to acquire complications, like a cloud shading from violet to steel blue. And Elena, for some reason, seemed to be holding him responsible for her opera ball being mentioned in the newspapers. But at least Samantha had come through with the goods, in the literary sense at any rate. Things might have been tolerable if she had been equally forthcoming in other departments.
But she was being hard to find, and, when found, hard to please. Several times he had parked his bicycle outside her basement flat at the appointed time in the early evening, only to discover that she was not at home, and with no note left on the door. More often, she was physically present but mentally absent, muttering vague generalisations about his lack of commitment to her. Employing a stratagem against theft which he had learned in New York, he would chain his bicycle to her area railings, loosen the wing nuts on the front wheel, carry the wheel down the steps and lean it against her wall before lying down with her to enjoy the only reliable privilege the English summer affords—the possibility of fornicating on the bed instead of in it. Even with her mind elsewhere she still had the body to make all the pedalling worth while, but things were not what they had been and he wondered how he might restore them to their former glory. He had planned to get out of going on the family holiday to Biarritz. Now that Charlotte was refusing to sleep with him he had every excuse for staying in London. Let David build the driftwood shelters. But Samantha announced that she would be going again, to the big house just down the coast where he had first met her. Once again she would be with her father, who disapproved of Lancelot, and her step-mother, who absolutely hated him, for trifling reasons arising from a too abruptly terminated affair twenty years before. But he would be able to meet Samantha in secret, or at any rate idolise her from a distance. He felt his need for her increasing as she slipped away. It was a pretty staggering thing for his need to do, when you considered the pitch of intensity it had long ago attained and never looked like losing.
Lancelot was still allowed into his house, but was required to lie down alone at all times. Charlotte had explained to him about David. Lancelot judged the embargo to be very reasonable but almost immediately began to feel hard done by. After all, it was not as if he were having an easy time with Samantha. He could have used some placid solidity to fall back on. Hints in this direction, however, were met with a certain impatience. He could come on the holiday if he wished, but he would have to sleep downstairs on the same floor as the children, even though David would not be there. Well, at least he would be able to keep up appearances.
The Windhovers' prospective house guests in Biarritz had not as yet come to any such crisis, but the easy days were gone. Sally had surprised herself by being able to go on sleeping with Nicholas: anyone who had dared tell her she would ever be capable of such a thing she would once have treated with contempt for the imputation as well as outrage at the intimacy. But the ructions which she was assured were going on between Victor and Elena left Victor little time to spare, so her time with him in New York still seemed like an interlude, and anyway even if she had wished to give Nicholas his cards she was not sure she knew how. He was so entertaining when he wasn't being the opposite, and he was only ever the opposite because of the intensity of his feelings. The worrying intensity of his feelings. Worrying because she didn't share them. This is how whores must feel. Although if they did, they wouldn't do it, would they? Let it all wait until after Biarritz. Victor would be in Salzburg or somewhere. The whole of London was breaking up for the hols, pip pip. Every living soul except ninety per cent of the population was getting ready to move out. Until then she went on enjoying Nicholas, while pretending to herself that she was not a hypocrite. It got harder to do that when she caught herself living for the telephone.
'You've got good nerves,' Victor said to her during one of his breakfast phone calls.
'I don't want to have good nerves,' she pointed out with some acerbity. He had the grace not to laugh.
He had nothing to laugh about, really. Elena had not taxed him directly on the subject but she didn't need to. Her intuition would have told her, even if his attentiveness had not given him away. Remembering her birthday had been a particularly glaring mistake. So against her standing orders, which forbade mentioning such a thing should it ever eventuate, he mentioned it.
'It's all your fault,' he said, in a foredoomed attempt to make light of it. 'You made me like the world.'
'If you see her in secret I might just about put up with this,' said Elena, with a serenity that boded ill. 'But if you acknowledge her even once, I'm gone for good. My pride wouldn't stand it. I'll just go back to Italy and preside over the final disintegration.'
All of which having been said and a lot more on the same lines, grief played its usual trick of repristinating passion. But the way the tears taste tells you what that's worth. She blamed life, not him, but he would rather have taken the blame himself. When they weren't together they were on the telephone far into the night. Elena withdrew from social life and the word went out that she was indisposed. But she and Victor had made arrangements to be in Salzburg together and felt that they might as well go.
So the exodus began. David and Gaga went furthest, to an Indian lake where they took up residence on a raft which had a little colour-washed plaster and lath replica of the Brighton Pavilion balanced on it to keep off the sun. At Gaga's request they had gone there in search of their first simplicity, but they could not find it, because when you have to look it's too late. In a by now desperate flight from capitalist values, the beautiful Gaga submitted herself gladly to her surroundings. She willed the primordial listlessness of the place to enter into her. Unfortunately some form of amoebic dysentery entered into her as well, and their holiday became a nightmare best left undocumented. The newspaper which had sponsored the trip received from David an article which could have gone equally well in a medical section as in a travel supplement. The editors could not make their minds up, so they spiked it.
Dick Toole and Delilah went to a Caribbean Island where one of the British princesses was taking a holiday during pregnancy. They had to come ashore at night in a rubber boat and found themselves outdone in terms of aggression by the French, German and Italian photographers, who had built hides in the undergrowth, disguised themselves as palm trees and were regularly delivered, wrapped up as parcels, on the verandah of the royal villa. But Dick Toole and Delilah picked up a lot of useful information by using Delilah's contacts and getting a few hangers-on to talk off the record in confidence. And the climate was so benign that even Dick Toole stripped down all the way to his grey vest, while together he and Delilah found several interesting new uses for her hired Scuba gear.
Charlotte, Lancelot and the children left for Biarritz in the weekly Caravelle. When the children had been much younger the annual trip had had some point, with even Lancelot glad to be away from some importunate innamorata. Now that the children were grown up there was less reason for going, in his opinion. The children still liked it because the town had all the same discos and Space Invaders machines as London but there was an even greater opportunity to spend their father's money. Apart from that the only real advantage was not to waste the house which by some old arrangement they took every year. It was poised on one of the few stretches of cliff above the Côte des Basques which showed as yet no signs of collapsing into the sea. Further south there was another beach on which Samantha annually appeared like a Homeric nymph, and then further south than that was Spain, represented, on a clear day, by mountains as delicately blue as the veins in a girl's breast. When the weather changed it gave you bad dreams, but when the sun shone it was a dream pure and simple.
Sally and Nicholas came down in the Porsche, stopping off when they felt like it and so taking two days for the trip. It should have been bliss and some of it even was. Nicholas loved it when Sally put her foot right down and the poplars lining the road turned into a picket fence. He loved hearing her talk French even if the waiters loved it too, not to mention the proprietors and half the other men in town, who would suddenly drop in to stage a spontaneous gesticulation competition. But something was going wrong in the nights. Until he had met her, he would have sworn that the body was the incarnation of the self. Looking at her as she lay stretched out, it was hard to believe that if she could give you all that then there was anything left to hold back. But she would not say the words.
In Biarritz she still wouldn't say them. Walking beside her on the beach gave him a hint of what it must have felt like to be Alexander the Great Sod or one of those characters. Frogs sucked in their stomachs and walked into each other. There were a lot of terrible Australian media types reputedly in search of a perfect wave that would remind them of the beach back home at Bang-Bang or Kogra-Wogra: the kind of Australians who carried a lot of food around with them just so they could talk with their mouths full. They moaned low as they stared up with crinkled slit eyes from under the brims of rat-eaten straw hats, saying things like 'Stone the crows'. It should have been ideal. He was friends again with Lancelot, whose children were less terrifying to look at without their bells and spangles, and whose wife looked happier than he had ever known her. Nicholas had the queen of the beach on his arm. In the cliff-top restaurant at sunset, with the sky the colour of faded pink geranium petals, her peach-skin tan was framed against the pale green tamarisks and the silver water like a meal for the eyes. But in the ground floor separate flat which they had been assigned, she would never say the proper things during the night. It was torture. At least Lancelot could be certain that Charlotte was all his, even to the extent of his being able to fix her up with a handbag holder like David. What was Lancelot's secret?
Only Charlotte knew exactly where she was. Usually she was under the driftwood shelter, reading up for a series of lectures on Burns. What a joyful spirit he had. What a man he must have been. She could see the idea of him now. That rhythm of his was something she now recognised as the pulse of her own bloodstream. It caused a stir under the cutty sark. Sometimes she almost felt like asking Lancelot into her bed again, but it would have spoiled things. All she had to do was wait. What a change, to know a man whose promises were worth something.
Lancelot suffered. He did not mind so much being barred from Charlotte's bed, as long as nobody knew. But the obvious happiness of Sally and Nicholas was more than he could stand. On the day after Samantha was supposed to arrive he rang her house and got her father, so he said 'Sorry, wrong number' in falsetto French and hung up, rather abashed that in his haste he had translated these words literally instead of using an idiomatic phrase. The next morning he rang again and got her, but all she could say was that it was difficult. So that afternoon, and all the afternoons that followed, he took the long walk south, hoping to catch a glimpse of her. Eventually he did, and she was with the actor. Everyone in the vicinity was staring towards that smile. Some of them were taking photographs. It was a topless beach and the sight of her digging her hard bare breasts into that man[1] was transmitted to Lancelot as the stab of a spear under his own heart. He looked wistfully across the rippling shallow water[2] which at mid-tide divided the two beaches and decided not to cross. He felt himself to be an ashen-faced Aschenbach watching Tadzio outlined against the sunset, or the hero of La Dolce Vita straining, against the sun's glare and his accumulated hangover, to keep the young girl — his picture of innocence — in focus[3]. Except that this one was not innocent. But what did he care about that? She saw him, waved him towards her, and when he did not move she came swayingly stepping, dragging her toes, towards him through the ankle-deep water, as once Nausikaa must have walked towards Odysseus where he lay face down in the foam, so that when he looked up through salted lashes he saw her as a curved shadow in the sunburst.
'Don't be a wet fart. Come and talk to us.'
'I can't.'
'Ring me in London then.'
Each day after that he would make the same journey, but lay in the shelter of the rocks like a frogman spying out an invasion beach. He watched them in secret. There didn't seem to be much going on. The actor spent a lot of his time floating on his back. Lancelot prayed that they would not walk north and they never did, or at least not until Sally and Nicholas were gone.
Sally went first. While Nicholas and Lancelot were down showing off to each other in the morning surf she said her goodbyes to Charlotte.
'I thought there was something wrong,' said Charlotte. 'Is it anything you can talk about?'
'Not really,' said Sally. 'I can't thank you enough, you know. It's been perfect from every aspect, except what's going on in my own thick head.'
'Someone you miss?'
'Something like that.'
'I know.'
'I hope I don't saddle you with a glum Nicholas.'
'We'll look after him.'
Sally loaded her bag into the back seat of the car. When the men came up to lunch she took Nicholas for a walk in the market and explained, or rather failed to explain. He wanted to come too but she wouldn't let him. When she was gone he explained how pressure of work had been haunting her. Two days later, after a record number of fruitless telephone calls, he got the weekly plane to London. But if he had gone straight away by train, or hired a car, he wouldn't have been back much sooner.
Lancelot should have left too but if he was going to be haunted he might as well stay near the ghost. She never came walking towards him along the Côte des Basques as she had done the year before and the blessed year before that, but on the second last night of their stay he saw her in the town centre. Having drunk everything in the house during and after dinner, he had staggered along the cliff path into town, there to find further sustenance. The lights and music in the old fishing port reminded him that it was the night of the Grand Bal. In the town centre there were thousands of people all crowded together and cheering the procession. Floats covered with crêpe flowers, coloured lights and pretty girls came swaying through. One of these was a particularly clever pastiche biplane made of paper flowers, with girl wing-walkers in white one-piece swimming costumes and goggled flying helmets. The central wing-walker, the one with the most beautiful figure, was Samantha. There was a big cheer for her and an even bigger one when the crowd saw that the smiling actor was the pilot. With his leather helmet and silk scarf he looked the part. He was always going to look the part. Lancelot wanted to go home.
So did Elena, oddly enough. After only a few days in Salzburg she announced that the ball had provided enough operatic experience for a lifetime and she could stand no more. So in the cliff-side house he took each year Victor was left to play solitary host while guests arrived and departed, sat down for grand meals, sallied out in blasé parties to attend the Festspielhaus, or toned their often ancient muscles by swimming against the artificial current of the indoor swimming pool. It would have been a good season, too. The Wittgenstein princesses never looked lovelier than they did that year, all sitting in a row for the first night of La Clemenza di Tito[4] in the Felsenreitschule. Wini Coburg died during the performance and nobody realised until after the final curtain, when it was noticed by the house manager that she was still sitting there after everybody else had gone off to a late dinner at the Hirsch. So she had lost her earring for the last time.
There was a lot going on but the festival felt empty without Elena, even though she had plainly been less than herself. Even Gus Disting could see that. He arrived in the company of the Japanese model, who was very keen to see Araberra and The Magic Fruit. Victor installed them in the summer house at the top of the garden.
'Vot's wrong with Elena, looking so sad?' Gus Disting asked Victor privately. 'A vooman in the prime of loveliness, it's not right.' Victor couldn't say, and not just because he was flabbergasted as usual by Gus Disting's caricature of an accent.
He couldn't say because what was happening was a secret even from themselves. They did not discuss it. Elena, who valued self-control above everything, would rather have died than utter a reproach. But when they were alone together she fell silent[5]. For the first time in many years she was taking things to make her sleep. They didn't make her sleep but robbed her of the quality she prized most — the capacity to mask her feelings. Victor sat on the edge of her bed looking useless, a position which did not become him. So she packed her bags and went home. All the way home.
From Milan the Settebello took her to Florence and at Florence she was met by a nondescript car driven by Sandro, almost the last of the old staff. Nobody owned a grand car any more, because of the kidnappings. Stopping several times for the provisions on his list, Sandro, whom she was saddened to find so very vague, took her out beyond Poggio a Caiano to a brown hill patched with olive groves and cypresses. The villas in that area are set within stuccoed walls[6], like the outworks of little fortresses. A single villa was all that was left of their once large estates. There had been a time when you could have looked out from the loggia and seen, with your gaze directed by avenues of cypresses, nothing but land belonging to the family. Now the avenues of cypresses led your gaze to the dye works, the light-bulb factory and the warehouse for blue plastic crates. But around the last house there were still some orchards and terraced vineyards, and there was no denying that her father still looked the part, with due allowance for all the doddering and shuffling. He received her in the library. They embraced, although from her side it might as well have been a formal hand-shake.
'Everyone says it was a great success.'
'It was, I have to admit. Your money was well spent, don't worry.'
'They say you and the cicisbeo dance cheek to cheek now.'
It was not a very profitable conversation. Why had she come? The heat was stunning.
Her sister was out in the fields and it was too hot to go looking, so she went upstairs to lie down in her old room. Probably that was why she had come. On the way there she looked into her father's study, where he kept what he called his real books. Complete leather-bound and gold-embossed editions of Gentile's philosophy and Mussolini's speeches: dross got up as treasure. The little marble bust of Mussolini was still there, with its silver eyes that lit up in oblique light. What a miracle that the Communists had never got in here and ransacked the place. Instead they were proving themselves model citizens by the way they administered all the surrounding towns. But no revolution had been necessary to kill all of this off. It had just died gradually of exhaustion. She was relieved to find herself so unashamed. There was nothing left to be ashamed of. Her father had remained loyal to his cause until the end. In Italian there is an expression for it: until the last day. People they knew had died on the quarry staircase at Mauthausen while her father continued with his scholastic researches into pure Latinity. He was still at it; loyal until the last day and beyond; proud, upright, kind, gentle and as mad as a hatter.
She slept fitfully but at least she slept. When her sister came in from the terrace they sat together over cold drinks in the tile-floored cool of the lobby where they had once raced tortoises in lanes made of books.
'You should have been there,' said Elena. 'The boy carried it off well.'
'I know,' said her sister. 'But there was so much to do here. The photographs were splendid.'
'Do you never get away at all?'
'I can't leave him for long or he'll be stumbling around shouting in his old uniform. And I can't be with people any more.'
'Will you come to the wedding?'
'I don't see how I can.'
'Perhaps that's what I should do,' said Elena reflectively.
'What?'
'Retreat from the world.'
'You? You must be very unhappy to say that.'
'Do I look it?'
'To someone who knows you.'
'It will pass.' But until it did it was killing her, so she poured it all out. So that was why she was there.
When the next President of the United States rang from Chicago the following morning she already felt, if not precisely better, at least alive enough to make a date with him for lunch in Paris early the week after next, and when Victor rang from Salzburg soon afterwards she made a point of letting him know. Then she went out into the vineyards with her sister and didn't talk to anyone else for days. It was a bit like talking to your reflection but at least your reflection didn't want anything. At first it seemed as if there were scorpions everywhere but later she noticed them less and with a length of strong wire bent into a hook she got the water flowing again from the lead pipe that lay hidden in the laughing mouth of the little green bronze Triton in the grotto — the Triton that might or might not have been by Giambologna. Water flowed from the peaches too, and the nectarines and the black grapes.
Read on: Chapter Twenty-Five


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NY attorney general sues Donald Trump, 3 kids for alleged business fraud
by: Mira Wassef, Associated Press, Ben Bradley
Updated: Sep 21, 2022 / 10:32 PM CDT
NEW YORK (PIX11) — New York Attorney General Letitia James is suing former President Donald Trump and three of his adult children for alleged business fraud.
James said Trump, Donald Trump Jr., Eric Trump, Ivanka Trump, and his company falsely inflated his net worth by billions of dollars to get banks to give him favorable interest rates. James is seeking to bar the former president and his children from ever running a company in New York again. She also wants Trump to pay back the $250 million he received.
"This investigation revealed that Donald Trump engaged in years of illegal conduct to inflate his net worth, to deceive banks and the people of the great state of New York," James said at the news conference. "Claiming you have money that you do not have does not amount to the art of the deal. It's the art of the steal."
James said there are 200 examples of alleged fraud in the complaint, including declaring his home was three times bigger than it was. She is also referring the matter to the federal authorities to conduct a criminal investigation.
The alleged criminal violations are falsifying business records, issuing false financial statements, insurance fraud, conspiracy, and bank fraud, James said. The case will also be referred to the Internal Revenue Service.
"They all should be held accountable," James said. "No one is above the law."
Last month, Trump invoked his Fifth Amendment right not to answer questions during a deposition related to James' investigation.
Trump's attorney, Alina Habba, and New York Republican Party Chairman Nick Langworthy blasted James' lawsuit.
Langworthy called it "one of the most brazen political publicity stunts I have seen during my lifetime." Habba said the lawsuit is "neither focused on the facts nor the law— rather, it is solely focused on advancing the attorney general's political agenda."
"It is abundantly clear that the Attorney General's Office has exceeded its statutory authority by prying into transactions where absolutely no wrongdoing has taken place," Habba said.
Longtime company executives Allen Weisselberg and Jeffrey McConney are also named defendants in the suit filed in state court Wednesday. Weisselberg, a senior Trump Organization adviser and formerly the company's longtime chief financial officer, has pleaded guilty to evading taxes and other charges.
Weisselberg admitted taking in over $1.7 million worth of untaxed extras — including school tuition for his grandchildren, free rent for a Manhattan apartment, and lease payments for a luxury car — and explicitly keeping some of the plums off the books.
James is looking to permanently bar Weisselberg and McConney from working as financial heads of any New York companies.
Snow, ice to blame in multi-vehicle crash near Kenosha
Chicago Daily Almanac


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AF, Navy join forces to protect
by adriennek • September 11, 2015 10:09 am - Updated April 25, 2022 10:36 am
Following force reductions in 2014, career fields around the Air Force likely experienced manpower strains in daily operations – the 56th Security Forces Squadron was no exception to these cuts.
In September 2013, the squadron was authorized 185 active-duty members. By October 2014, the unit was reduced to a mere 136 active-duty members.
With these reductions, management searched for solutions to offset the manpower shortage, yet still have the ability to maintain security of the installation and continue to allow installation gates to remain open to the community. Some immediate solutions were the reassigning of administrative jobs and redistributing additional duties to other personnel in the unit. However, more was needed to align the right posture for security. This was when the concept of joining forces with our Navy counterparts at Luke came to light.
The security forces leadership team met with the Navy master of arms leadership team at the Naval Operations Support Center and discussed the possibility of joining forces. The teams developed the idea and then drafted the first-ever joint security posting memorandum of agreement. The agreement consisted of training, equipping and certifying the master of arms personnel to enable them to provide base security services to the community, thus adding additional manpower to daily operations. By the beginning of 2015, the concept of a joint security force at Luke became a reality.
In the past few months, you may have notice Navy personnel posted with security forces members. If you attended the NFL Pro Bowl practice or the VH1 concert, you likely noticed them standing guard for the events. In February, the Integrated Medical Services Marathon took place on the installation and our Navy partners assisted with security for the runners. In June, Navy master of arms personnel stood post at the South Gate directing traffic onto the installation during their drill weekend and attended combat arms training, qualifying on their firearms.
"We are extremely fortunate to have the master of arms contribute to the mission at Luke," said 1st Lt. Jonathan Warzeka, 56th SFS operations officer. "Together, the units have worked tirelessly to make this joint effort happen."
Be on the lookout for our Navy partners on post as they will join security forces personnel during their next drill weekend Sept. 25 through 27.
Tags: 56th Security Forces Squadron