\section{Architecture}
\begin{figure}[htb]
\includegraphics[width=\columnwidth] {framework.png}
\caption{\label{fig:framework} Illusion of the system of DCAF. \textit{Request Value Online Estimation} module will score each request conditioned on action $j$ through online features of which estimator is trained offline. \textit{Policy Execution} module mainly takes charge of executing the final action $j$ for each request based on the system status collected by \textit{Information Collection and Monitoring} module, $\lambda$ calculated offline and $Q_{ij}$ obtained from previous module.}
\medskip
\end{figure}
In general, DCAF is comprised of online decision maker and offline estimator:
\begin{itemize}
    \item The online modules make the final decision based on personalized request value and system status.
    \item The offline modules leverage the logs to calculate the Lagrange Multiplier $\lambda$ and train a estimator for the request expected value conditioned on actions based on historical data.
\end{itemize}
\subsection{Online Decision Maker}
\subsubsection{Information Collection and Monitoring}~\\
This module monitors and provides timely information about the system current status which includes GPU-utils, CPU-utils, runtime (RT), failure rate, and etc. The acquired information enables the framework to dynamically allocate the computation resource without exceeding the budget by limiting the action space.
\subsubsection{Request Value Estimation}~\\
This module estimates the request's $Q_{ij}$ based on the features provided in information collection module. Notably, to avoid growing the system load, the online estimator need to be light-weighted, which necessitates the balance between efficiency and accuracy. One possible solution is that the estimation of $Q_{ij}$ should re-utilize the request context features adequately, e.g. high-level features generated by other models in different modules.
\subsubsection{Policy Execution}~\\
Basically, this module assigns the best action $j$ to request $i$ by Equation (6). Moreover, for the stability of online system, we put forward a concept called \textit{MaxPower} which is an upper bound for $q_j$  to which each request must subject. DCAF sets a limit on the \textit{MaxPower} in order to strongly control the online engine. The \textit{MaxPower} is automatically controlled by system's runtime and failure rate through control loop feedback mechanism, e.g. Proportional Integral Derivative (PID) \cite{ang2005pid}. The introduction of \textit{MaxPower} guarantees that the system can adjust itself and remain stable automatically and timely when encountering sudden request spikes. \newline
According to the formulation of PID, $u(t)$ and $e(t)$ are the control action and system unstablity at time step $t$. For $e(t)$, we define it as the weighted sum of average runtime and fail rates over a time interval which are denoted by $rt$ and $fr$ respectively. $k_p$, $k_i$ and $k_d$ are the corresponding weights for  proportional,integral and derivative control. ${\theta}$ means a tuned scale factor for the weighted sum of $rt$ and $fr$. Formally,
\begin{align}
u(t)= k_p{e(t)}+k_i{\sum_{n=1}^{t}e(t)}+k_d({e(t)-e(t-1)})
\end{align}

\begin{algorithm}
\caption{PID Control for MaxPower}
\begin{flushleft}
1: \textbf{Input:} $k_p$, $k_i$, $k_{d}$, $MaxPower$ \\
2: \textbf{Output:} $MaxPower$ \\
3: \textbf{while} (true): \\
4: \hspace{5mm} Obtain $rt$ and $fr$ from \textit{Information Collection and Monitoring} \\
5: \hspace{5mm} ${e(t)={rt}+\theta{fr}}$ \\
6: \hspace{5mm} $u(t)= k_p{e(t)}+k_i{\sum_{n=1}^{t}e(t)}+k_d({e(t)-e(t-1)})$\\
7: \hspace{5mm} Update $MaxPower$ with $u(t)$\\
8: \textbf{end while} \\
\end{flushleft}
\end{algorithm}

\subsection{Offline Estimator}
\subsubsection{Lagrange Multiplier Solver} ~\\
As mentioned above, we could get the global optimal solution of the Lagrange Multipliers by a simple bisection search method. In real case, we take logs as a request pool to search a best candidate Lagrange Multiplier $\lambda$. Formally,
\begin{itemize}
    \item Sample $N$ records from the logs with $Q_{ij}$, $q_j$ and computation cost $C$, e.g. the total amount of advertisements that request the CTR model within a time interval.
    \item Adjust the computation cost $C$ by the current system status in order to keep the dynamic allocation problem under constraint in time. For example, we denote regular QPS by $QPS_r$ and current QPS by $QPS_c$. Then the adjusted computation cost $\hat{C}$ is $C \times \nicefrac{QPS_r}{QPS_c}$, which could keep the $N$ records under the current computation constraint.
    \item Search the best candidate Lagrange Multiplier $\lambda$ by Algorithm (1)
\end{itemize}
It's worth noting that we actually assume the distribution of the request pool is the same as online requests, which could probably introduce the bias for estimating Lagrange Multiplier. However, in practice, we could partly remove the bias by updating the $\lambda$ frequently.
\subsubsection{Expected Gain Estimator} ~\\
In our settings, for each request,  $Q_{ij}$ is associated with eCPM under different action $j$ which is the common choice for performance metric in the field of online display advertising. Further, we build a CTR model to estimate the CTR, because the eCPM could be decomposed into $ctr \times bid$ where the bids are usually provided by advertisers directly. It is notable that the CTR model is conditioned on actions in our case, where it is essential to evaluate each request gain under different actions.
And this estimator is updated routinely and provides real-time inference in \textit{Policy Execution} module.

\section{Conclusion}
In this paper, we propose a noval dynamic computation allocation framework (DCAF), which can break pre-defined quota constraints within different modules in existing cascade system. By deploying DCAF online, we empirically show that DCAF consistently maintains the same performance with great computation resource reduction in online advertising system, and meanwhile, keeps the system stable when facing the sudden spike of requests. Specifically, we formulate the dynamic computational allocation problem as a knapsack problem. Then we theoretically prove that the total revenue can be maximized under a computation budget constraint by properly allocating resource according to the value of individual request. Moreover, under some general assumptions, the global optimal Lagrange Multiplier $\lambda$ can also be obtained which finally completes the constrained optimization problem in theory. Moreover, we put forward a concept called \textit{MaxPower}  which is controlled by a designed control loop feedback mechanism in real-time. Through \textit{MaxPower} which imposes constraints on the range of action candidates, the system could be controlled powerfully and automatically.
\section{experiments}
\subsection{Offline Experiments}
For validating the framework's correctness and effectiveness, we extensively analyse the real-world logs collected from the display advertising system of Taobao and conduct offline experiments on it. As mentioned above, it is common practice for most systems to ignore the differences in value of requests and execute same procedure on each request. Therefore, we set equally sharing the computation budget among different requests as the \textbf{baseline} strategy.
As shown in Figure \ref{fig:cascade}, we simulate the performance of DCAF in Ranking stage by offline logs. In advance, we make it clear that all data has been rescaled to avoid breaches of sensitive commercial data. We conduct our offline and online experiments in Taobao's display advertisement system where we spend the GPU resource automatically through DCAF. In detail, we instantiate the dynamic allocation problem as follow:
\begin{itemize}
    \item Action $j$ controls the number of advertisements that need to be evaluated by online CTR model in Ranking stage.
    \item $q_j$ represents the advertisement's quota for requesting the online CTR model.
    \item $Q_{ij}$ is the sum of top-k ad's eCPM for request $i$ conditioned on action $j$ in Ranking stage which is equivalent to online performance closely. And $Q_{ij}$ is estimated in the experiment.
    \item $C$ stands for the total number of advertisements that are requesting online CTR model in a period of time within the serving capacity.
    \item \textbf{Baseline}: The original system, which allocates the same computation resource to different requests. With the baseline strategy, system scores the same number of advertisements in Ranking stage for each request.
\end{itemize}

\begin{figure}[htb]
\includegraphics[width=\columnwidth] {Figure_1.png}
\caption{\label{fig:exp1} Global optima under different $\lambda$ candidates. In Figure \ref{fig:exp1}, x-axis stands for $\lambda$'s candidate; left y-axis represents $\sum_{ij}{x_{ij}Q_{ij}}$; right-axis denotes the corresponding cost. The red shadow area corresponds to the exceeding part of $max\sum_{ij}{x_{ij}Q_{ij}}$ beyond the baseline. And yellow shadow area is the reduction of $\sum_{ij}{x_{ij}q_{j}}$ under these $\lambda$'s compared with the baseline. Random strategy is also shown in Figure \ref{fig:exp1} for comparison with DCAF.}
\medskip
\end{figure}

\textbf{Global optima under different $\lambda$ candidates. } In DCAF, the Lagrange Multiplier $\lambda$ works by imposing constraint on the computation budget. Figure \ref{fig:exp1} shows the relation among $\lambda$'s magnitude, $max\sum_{ij}{x_{ij}Q_{ij}}$ and its corresponding $\sum_{ij}{x_{ij}q_{j}}$ under fixed budget constraint. Clearly, $\lambda$ could monotonically impact on both  $max\sum_{ij}{x_{ij}Q_{ij}}$ and its corresponding $\sum_{ij}{x_{ij}q_{j}}$. And it shows that the DCAF outperforms the baseline when $\lambda$ locates in an appropriate interval. As demonstrated by the two dotted lines, in comparison with the baseline, DCAF can achieve both higher performance with same computation budget and same performance with much less computation budget. Compared with random strategy, DCAF's performance outmatches the random strategy's to a large extent.   \newline

\begin{figure}[htb]
\includegraphics[width=\columnwidth] {Figure_2.png}
\caption{\label{fig:exp2} Comparison of DCAF with the original system on computation cost. In Figure \ref{fig:exp2}, x-axis denotes the $\sum_{ij}{x_{ij}Q_{ij}}$; y-axis represents the$\sum_{ij}{x_{ij}q_{j}}$. For points on the two lines with same x-coordinate, Figure \ref{fig:exp2} shows that DCAF always perform as well as the baseline by much less computation resource.}
\medskip
\end{figure}

\textbf{Comparison of DCAF with the original system on computation cost. } Figure \ref{fig:exp2} shows that DCAF consistently accomplish same performance as the baseline and save the cost by a huge margin. Furthermore, DCAF plays much more important role in more resource-constrained systems.

\begin{figure}[htb]
\includegraphics[width=\columnwidth] {Figure_3.png}
\caption{\label{fig:exp3} Total eCPM and its cost over different actions. In this figure, x-axis stands for action $j$'s and left y-axis represents $\sum_{ij}{x_{ij}Q_{ij}}$ conditioned on action $j$; right-axis denotes the corresponding cost. For each action $j$, we sum over $Q_{ij}$ which is the the sum of top-k ad's eCPM for requests that are assigned action $j$ by DCAF.}
\medskip
\end{figure}

\textbf{Total eCPM and its cost over different actions. } As shown by the distributions in Figure \ref{fig:exp3}, we could see that DCAF treats each request differently by taking different action $j$. And $\nicefrac{\sum_{ij}Q_{ij}}{\sum_{ij}q_{j}}$ is decreasing with action $j$'s which empirically show that the relation between expected gain and its corresponding cost follows the law of diminishing marginal utility in total.

\subsection{Online Experiments}
DCAF is deployed in Alibaba display advertising system since 2020. From 2020-05-20 to 2020-05-30, we conduct online A/B testing experiment to validate the effectiveness of DCAF. The settings of online experiments are almost identical to offline experiments. Action $j$ controls the number of advertisements for requesting the CTR model in Ranking stage. And we use a simple linear model to estimate the $Q_{ij}$. The original system without DCAF is set as baseline. The DCAF is deployed between Pre-Ranking stage and Ranking stage which is aimed at dynamically allocating the GPU resource consumed by Ranking's CTR model. Table \ref{exp:samecost} shows that DCAF could bring improvement while using the same computation cost. Considering the massive daily traffic of Taobao, we deploy DCAF to reduce the computation cost while not hurting the revenue of the ads system. The results are illustrated in Table \ref{exp:samerpm}, and DCAF reduces the computation cost with respect to the total amount of advertisements requesting CTR model by 25\% and total utilities of GPU resource by 20\%. It should be noticed that, in online system, the $Q_{ij}$ is estimated by a simple linear model which may be not sufficiently complex to fully capture data distribution. Thus the improvement of DCAF in online system is less than the results of offline experiments. This simple method enables us to demonstrate the effectiveness of the overall framework which is our main concern in this paper. In the future, we will dedicate more efforts in modeling $Q_{ij}$. Figure \ref{fig:exp4} shows the performance of DCAF under the pressures of online traffic in extreme case e.g. Double 11 shopping festival. By the control mechanism of \textit{MaxPower}, the online serving system can react to the sudden rising of traffic quickly, and make the system back to normal status by consistently keeping the fail rate and runtime at a low level. It is worth noticing that the control mechanism of \textit{MaxPower} is superior to human interventions in the scenario that the large traffic arrives suddenly and human interventions inevitably delay.

\begin{table}[!h]
  \caption{Results with Same Computation Budget}
\label{exp:samecost}
  \begin{tabular}{lcl}
    \toprule
      & CTR & RPM\\
    \midrule
 Baseline & +0.00\% & +0.00\% \\
	DCAF & +0.91\% & +0.42\% \\
  \bottomrule
\end{tabular}
\end{table}

\begin{table}[!h]
  \caption{Results with Same Revenue}
\label{exp:samerpm}
  \begin{tabular}{lcccl}
    \toprule
      & CTR & RPM & Computation Cost & GPU-utils\\
    \midrule
 Baseline & +0.00\% & +0.00\% & -0.00\% & -0.00\%\\
	DCAF & -0.57\% & +0.24\% & -25\% & -20\%\\
  \bottomrule
\end{tabular}
\end{table}

\begin{figure}[htb]
\centering
\begin{subfigure}[b]{\columnwidth}
   \includegraphics[width=1\linewidth]{Figure_4.png}
   \caption{}
   \label{fig:Ng1}
\end{subfigure}
\begin{subfigure}[b]{\columnwidth}
   \includegraphics[width=1\linewidth]{Figure_5.png}
   \caption{}
   \label{fig:Ng2}
\end{subfigure}
\caption{\label{fig:exp4} The effect of \textit{MaxPower} mechanism. In this experiments, we manually change the traffic of system at time $158$ and the requests per second increase 8-fold. Figure \ref{fig:Ng1} shows the trend of \textit{MaxPower} over time and Figure \ref{fig:Ng2} shows the trend of fail rate over time. As shown in Figure \ref{fig:exp4}, the \textit{MaxPower} takes effect immediately when the QPS is rising suddenly which makes the fail rate keep at a lower level. At the same time, the base strategy fails to serve some requests, because it does not change the computing strategy while the computation power of system is insufficient.}
\medskip
\end{figure}
\section{Formulation}
We formulate the dynamic computation allocation problem as a knapsack problem which is aimed at maximizing the total revenue under the computation budget constraint. We assume that there are $N$ requests $\{i=1,\dots,N\}$ requesting the e-commerce platform within a time period. For each request, $M$ actions $\{j=1,\dots,M\}$ can be taken. We define  $Q_{ij}$ and $q_j$ as the expected gain for request $i$ that is assigned action $j$ and the cost for action $j$ respectively. $C$ represents the total computation budget constraint within a time period. For
instance, in the display advertising system deployed in e-commerce, $q_j$ usually stands for items (ads) quota that request the online engine to evaluate, which positively correlate with system load in usual. And $Q_{ij}$ usually represent the eCPM~(effective cost per mille) conditioned on action $j$ which directly proportional to  $q_j$. $x_{ij}$ is the indicator that request $i$ is assigned action $j$. For each request, there is one and only one action $j$ can be taken, in other words, $x_{i.}$ is an one-hot vector. \newline
Following the definitions above, for each request, our target is to maximize the total revenue under computation budget by assigning each request $i$ with appropriate action $j$. Formally,
\begin{align*}
{\rm \max_j}&\sum_{ij}{x_{ij}Q_{ij}} \\
     &{\rm s.t.} \sum_{ij}{x_{ij}q_{j}} \le C \\
     &\sum_j{x_{ij}} \le 1 \\
     &x_{ij} \in \{0,1\} \numberthis
\end{align*}
where we assume that each individual request has its "personalized" value, thus should be treated differently. Besides, request expected gain is correlated with action $j$ which will be automatically taken by the platform in order to maximize the objective under the constraint. In this paper, we mainly focus on proving the effectiveness of DCAF's framework as a whole. However, in real case, we are faced with several challenges which are beyond the scope of this paper. We simply list them as below for considering in the future:
\begin{itemize}
    \item The dynamic allocation problem \cite{berger2018robinhood} are usually coupled with real-time request and system status. As the online traffic and system status are both varying with time, we should consider the knapsack problem to be real-time, e.g. real-time computation budget.
    \item $Q_{ij}$ are unknown, thus needs to be estimated. $Q_{ij}$ prediction is vital to maximize the objective, which means real-time and efficient approaches are required to estimate the value. Besides, to avoid increasing the system's burden, it is essential for us to consider light-weighted methods.
\end{itemize}

\section{Future Work}
Fairness has attracted more and more concerns in the fields of recommendation system and online display advertisements. In this paper we propose DCAF, which allocate the computation resource dynamically among requests. The values of request vary with time, scenario, users and other factors, that incite us to treats each request differently and customize the computation resource for it. But we also noticed that DCAF may discriminate among users. While the allocated computation budgets varying with users, DCAF may leave a impression that it would aggravate the unfairness phenomenon of system further. In our opinion, the unfair problem stems from that all the approaches to model users are data-driven. Meanwhile
most of systems create a data feedback loop that a system is trained and evaluated on the data impressed to users \cite{chaney2018algorithmic}. We think the fairness of recommender system and ads system is important and needs to be paid more attention to. In the future, we will analyse the long-term effect for fairness of DCAF extensively and include the consideration of it in DCAF carefully. \newline
Besides, DCAF is still in the early stage of development, where modules in the cascade system are considered independently and the action $j$ is defined as the number of candidate to be evaluated in our experiments. Obviously, DCAF could work with diverse actions, such as models with different calculation complexity. Meanwhile, instead of maximizing the total revenue in particular module, DCAF will achieve the global optima in the view of the whole cascade system in the future. Moreover, in the subsequent stages, we will endow DCAF with the abilities of quick adaption and fast reactions. These abilities will enable DCAF to exert its full effect in any scenario immediately.
\section{Introduction}

Modern large-scale systems such as recommender system and online advertising are built upon computation-intensive infrastructure \cite{cheng2016wide}  \cite{zhou2018deep} \cite{zhou2019deep}. With the popularity of e-commerce shopping, e-commerce platform such as Taobao, the world's leading e-commerce platforms,  are now enjoying a huge boom in traffic \cite{cardellini1999dynamic}, e.g. user requests at Taobao are increasing year by year. As a result, the system load is under great pressures \cite{zhou2018rocket}. Moreover, request fluctuation also gives critical challenge to online serving system. For example, the Taobao recommendation system always bears many spikes of requests during the period of Double 11 shopping festival.

\begin{figure}[thb]
\includegraphics[width=\columnwidth] {cascade.png}
\caption{\label{fig:cascade} Illustration of our cascaded display advertising system. Each request will be served through these modules sequentially. Considering the limitation of computation resource and latency for online serving system, the fixed quota of candidate advertisements, denoted by $N$ for each module, is usually pre-defined manually by experience. }
\medskip
\small
\end{figure}

To address the above challenges, the prevailing practices for online engine are: 1) decomposing the cascade system \cite{liu2017cascade} into multiple modules and manually allocating a fixed quota for each module by experience, as shown in Figure~\ref{fig:cascade}; 2) designing many computation downgrade plans in case the sudden traffic spikes arrive and manually executing these plans when needed.

These non-automated strategies are often lack of flexibility and require human interventions. Furthermore, most of these practices often impact on all requests once they are executed and ignore the fact that the value of requests varies greatly. Obviously it is a straightforward but better strategy to allocate the computation resource by biasing towards the requests that are more valuable than others, for maximizing the total revenue.

Considering the shortcomings of existing works, we aim at building a dynamic allocation framework that can allocate the computation budget flexibly among requests. Moreover, this framework should also take into account the stability of the online serving system which are frequently challenged by request boom and spike. Specifically, we formulate the problem as a knapsack problem, of which objective is to maximize the total revenue under a computation budget constraint. We propose a dynamic allocation framework named DCAF which could consider both computation budget allocation and stability of online serving system simultaneously and automatically.

Our main contributions are summarized as follow:

\begin{itemize}
    \item We break through the stereotypes in most cascaded systems where each individual module is limited by a static computation budget independently. We introduce a brand-new idea that computation budget can be allocated w.r.t the value of traffic requests in a "personalized" manner.
    \item  We propose a dynamic allocation framework DCAF which could guarantee in theory that the total revenue can be maximized under a computation budget constraint. Moreover, we provide an powerful control mechanism that could always keep the online serving system stable when encountering the sudden spike of requests.
    \item DCAF has been deployed in the display advertising system of Taobao, bringing a notable improvement. To be specific, system with DCAF maintains the same business performance with 20\% GPU (Graphics Processing Unit) resource reduction for online ranking system. Meanwhile, it greatly boosts the online engine's stability.
    \item By defining the new paradigm, DCAF lays the cornerstone for jointly optimizing the cascade system among different modules and raising the ceiling height for performance of online serving system further.
\end{itemize}

\section{Acknowledgment}
We thanks Zhenzhong Shen, Chi Zhang for helping us on deep neural net inference optimization and conducting the dynamic resource allocation experiments.

\bibliographystyle{ACM-Reference-Format}

\section{Methodology}
\subsection{Global Optimal Solution and Proof}
To solve the problem, we firstly construct the Lagrangian from the formulation above,
\begin{align*}
L = -\sum_{ij}{x_{ij}Q_{ij}}+\lambda (\sum_{ij}{x_{ij}q_{j}}-C)+\sum_i{(\mu_i(\sum_j{x_{ij}}-1))} \\
= \sum_{ij}{x_{ij}(-Q_{ij}+\lambda q_j+\mu_i)}-\lambda C-\sum_i{\mu_i} \\
{\rm s.t.} \lambda \ge 0 \\
\mu_i \ge 0 \\
x_{ij} \ge 0 \numberthis
\end{align*}
where we relax the discrete constraint for the indicator $x_{ij}$, we could show that the relaxation does no harm to the optimal solution. From the primal, the dual function \cite{boyd2004convex} is
\begin{align}
    {\rm \max_{\lambda,\mu}}\ {{\rm \min_{x_{ij}}}}(\sum_{ij}{x_{ij}(-Q_{ij}+\lambda q_j+\mu_i)}-\lambda C-\sum_i{\mu_i})
\end{align}
With $x_{ij} \ge 0$ ($x_{ij} \le 1$ is implicitly described in the Lagrangian), the linear function is bounded below only when $-Q_{ij}+\lambda q_j+\mu_i\ge0$. And only when $-Q_{ij}+\lambda q_j+\mu_i=0$, the inequality $x_{ij} > 0$ could hold which means $x_{ij} = 1$ in our case (remember that the $x_{i.}$ is an one-hot vector). Formally,
\begin{align*}
{\rm \max_{\lambda,\mu}}(-\lambda C-\sum_i{\mu_i}) \\
{\rm s.t.} -Q_{ij}+\lambda q_j+\mu_i\ge0 \\
\lambda \ge 0 \\
\mu_i \ge 0 \\
x_{ij} \ge 0 \numberthis
\end{align*}
As the dual objective is negatively correlated with $\mu$, the global optimal solution for $\mu$ would be
\begin{align}\label{equ5}
\mu_{i} = {\rm \max_{j}}(Q_{ij}-\lambda q_j)
\end{align}
Hence, the global optimal solution to $x_{ij}$ that indicate which action $j$ could be assigned to request $i$ is
\begin{align}
j = {\rm arg\max_{j}}(Q_{ij}-\lambda q_j)
\end{align}
From Slater’s theorem \cite{slater1950lagrange}, it can be easily shown that the Strong Duality holds in our case, which means that this solution is also the global optimal solution to the primal problem.

\subsection{Parameter Estimation}
\subsubsection{Lagrange Multiplier}~\\
The analytical form of Lagrange multiplier cannot be easily, or even possibly derived in our case. And meanwhile, the exact global optimal solution in arbitrary case is computationally prohibitive. However, under some general assumptions, simple bisection search could guarantee that the global optimal $\lambda$ could be obtained. Without loss of generality, we reset the indices of action space by following the ascending order of $q_j$'s magnitude.
\begin{assumption}\label{ass1}
$Q_{ij}$ is monotonically increasing with $j$.
\end{assumption}

\begin{assumption}\label{ass2}
$\nicefrac{Q_{ij}}{q_j}$ is monotonically decreasing with $j$.
\end{assumption}

\begin{lemma}\label{lem0}
Suppose Assumptions (\ref{ass1}) and (\ref{ass2}) hold, for each $i$, $\nicefrac{Q_{i{j_1}}}{q_{j_1}} \ge \nicefrac{Q_{i{j_2}}}{q_{j_2}}$ will hold if $\lambda_1 \ge \lambda_2$, where $j_1$ and $j_2$ are the actions that maximize the objective under $\lambda_1$ and $\lambda_2$ respectively.
\end{lemma}
\begin{proof}
As Equation (\ref{equ5}) and $\mu_{i}\ge0$, the inequality $Q_{ij}-\lambda q_j \ge 0$ holds. Equally, $\nicefrac{Q_{ij}}{q_j} \ge \lambda$ holds. Suppose $\nicefrac{Q_{i{j_1}}}{q_{j_1}} < \nicefrac{Q_{i{j_2}}}{q_{j_2}}$, we have $\nicefrac{Q_{i{j_2}}}{q_{j_2}} > \nicefrac{Q_{i{j_1}}}{q_{j_1}} \ge \lambda_1 \ge \lambda_2$. However, we could always find $j_2^*$ such that $Q_{i{j_2^*}} \ge Q_{i{j_2}}$ and  ${q_{j_2^*}} > {q_{j_2}}$ where $\nicefrac{Q_{i{j_1}}}{q_{j_1}} \ge \nicefrac{Q_{i{j_2^*}}}{q_{j_2^*}} \ge \lambda_1 \ge \lambda_2$ such that $Q_{i{j_2^*}} \ge Q_{i{j_2}}$  by following the Assumptions (\ref{ass1}) and (\ref{ass2}). In order words, $j_2$ is not the action that maximize the objective. Therefore, we have $\nicefrac{Q_{i{j_1}}}{q_{j_1}} \ge \nicefrac{Q_{i{j_2}}}{q_{j_2}}$.
\end{proof}

\begin{lemma}\label{lem1}
Suppose Assumptions (\ref{ass1}) and (\ref{ass2}) could be satisfied, both $max\sum_{ij}{x_{ij}Q_{ij}}$ and its corresponding $\sum_{ij}{x_{ij}q_{j}}$ are monotonically decreasing with $\lambda$.
\end{lemma}
\begin{proof}
With $\lambda$ increasing, $\nicefrac{Q_{ij}}{q_j}$ is also increasing monotonically by Lemma (\ref{lem0}). Moreover, by Assumptions (\ref{ass1}) and (\ref{ass2}), we conclude that both $max\sum_{ij}{x_{ij}Q_{ij}}$  and its corresponding $\sum_{ij}{x_{ij}q_{j}}$ are monotonically decreasing with $\lambda$.
\end{proof}

\begin{theorem}
Suppose Lemma (\ref{lem1}) holds, the global optimal Lagrange Multiplier $\lambda$ could be obtained by finding a solution that make $\sum_{ij}{x_{ij}q_{j}} = C$ hold through bisection search.
\end{theorem}
\begin{proof}
By Lemma (\ref{lem1}), this proof is almost trivial. We denote the Lagrange Multiplier that makes $\sum_{ij}{x_{ij}q_{j}} = C$ hold as $\lambda^*$. Obviously, the increase of $\lambda^*$ will result in computation overload and the decrease of $\lambda^*$ will inevitably reduce max$\sum_{ij}{x_{ij}Q_{ij}}$ due to the monotonicity in Lemma (\ref{lem1}). Hence, $\lambda^*$ is the global optimal solution to the constrained maximization problem. Besides, the bisection search must work in this case which is also guaranteed by the monotonicity.
\end{proof}
Assumption (\ref{ass1}) usually holds because the gain is directly proportional to the cost in general,e.g. more sophisticated models usually bring better online performance. For Assumption (\ref{ass2}), it follows the law of diminishing marginal utility \cite{scott1955fishery}, which is an economic phenomenon and reasonable in our constrained dynamic allocation case.

The algorithm for searching Lagrange Multiplier $\lambda$ is described in Algorithm \ref{algo:lagrange}. In general, we implement the bisection search over a pre-defined interval to find out the global optimal solution for $\lambda$. Suppose $\min_j\sum_{j}{q_{j}} \le C \le \max_j \sum_{j}{q_{j}}$ (o.w there is no need for dynamic allocation), it can be easily shown that $\lambda$ locates in the interval $[0, \min_{ij}(\nicefrac{Q_{ij}}{q_{j}})]$. Then we get the global optimal $\lambda$ through bisection search of which target is the solution of $\sum_{ij}{x_{ij}q_{j}} = C$.
\begin{algorithm}
\caption{\label{algo:lagrange} Calculate Lagrange Multiplier}
\begin{flushleft}
1: \textbf{Input:}  $Q_{ij}$, $q_j$, $C$, interval $[0, \min_{ij}(\frac{Q_{ij}}{q_{j}})]$ and tolerance $\epsilon$ \\
2: \textbf{Output:} Global optimal solution of Lagrange Multiplier $\lambda$\\
3: Set $\lambda_l = 0$, $\lambda_r = \min_{ij}(\frac{Q_{ij}}{q_{j}})$, $gap = +\infty$ \\
4: \textbf{while} ($gap > \epsilon$): \\
5: \hspace{5mm} $\lambda_m = \lambda_l + \frac{\lambda_r - \lambda_l}{2}$ \\
6: \hspace{5mm} Choose action $j_m^*$ by
\centerline{$\{j: {\rm arg\max_{j}}(Q_{ij}-\lambda_m q_j), Q_{ij}-\lambda_m q_j \ge 0\}$}\\
7: \hspace{5mm} Calculate the $\sum_i q_{j_l^*}$ denoted by $C_m$ \\
8: \hspace{5mm} $gap = |C_m - C|$ \\
9:  \hspace{5mm} \textbf{if} $gap \le \epsilon$: \\
10: \hspace{1cm} \textbf{return} $\lambda_m$ \\
11: \hspace{5mm} \textbf{else if}  $C_m \le C$: \\
12 : \hspace{1cm}  $\lambda_l = \lambda_m$ \\
13 : \hspace{5mm} \textbf{else}: \\
14:   \hspace{1cm}  $\lambda_r = \lambda_m$ \\
15: \textbf{end while} \\
16: \textbf{Return} the global optimal $\lambda_m$ which satisfies $|\sum_i q_{j_l^*} - C| \le \epsilon$.
\end{flushleft}
\end{algorithm}

For more general cases, more sophisticated method other than bisection search, e.g. reinforcement learning, will be conducted to explore the solution space and find out the global optimal $\lambda$.

\subsubsection{Request Expected Gain}~\\
In e-commerce, the expected gain is usually defined as online performance metric e.g. Effective Cost Per Mile (eCPM), which could directly indicate each individual request value with regard to the platform.
 Four categories of feature are mainly used: User Profile, User Behavior, Context and System status. It is worth noticing that our features are quite different from typical CTR model:
\begin{itemize}
    \item Specific target ad feature isn't provided because we estimate the CTR conditioned on actions.
    \item System status is included where we intend to establish the connection between system and actions.
    \item The context feature consists of the inference results from previous modules in order to re-utilize the request information efficiently.
\end{itemize}

\section{Related work}
Quite a lot of research have been focusing on improving the serving performance. Park et al. \cite{park2018deep} describes practice of serving deep learning models in Facebook. Clipper \cite{crankshaw2017clipper} is a general-purpose low-latency prediction serving system. Both of them use latency, accuracy, and throughput as the optimization target of the system. They also mentioned techniques like batching, caching, hyper-parameters tuning, model selection, computation kernel optimization to improve overall serving performance. Also, many research and system use model compression \cite{han2015deep}, mix-precision inference, quantization \cite{gupta2015deep,courbariaux2015binaryconnect}, kernel fusion \cite{chen2018tvm}, model distillation \cite{hinton2015distilling, zhou2018rocket} to accelerate deep neural net inference.

Traditional work usually focus on improving the performance of individual blocks, and the overall serving performance across all possible queries. Some new systems have been designed to take query diversity into consideration and provide dynamic planning. DQBarge \cite{chow2016dqbarge} is a proactive system using monitoring data to make data quality tradeoffs. RobinHood \cite{berger2018robinhood} provides tail Latency aware caching to dynamically allocate cache resources. Zhang et al. \cite{zhang2019e2e} takes user heterogeneity into account to improve quality of experience on the web. Those systems provide inspiring insight into our design, but existing systems did not provide solutions for computation resource reduction and comprehensive study of personalized planning algorithms.


\section{Introduction}
Formation of relativistic jets in active galactic nuclei (AGN) is one of the
biggest challenges in astrophysics. The radio galaxy M87 is one of the closest
examples of this phenomena, and its jet has been investigated across the entire
electromagnetic spectrum over years~\citep[e.g.,][]{owen1989, biretta1999,
harris2006, abramowski2012}. Due to its proximty
\citep[$D=16.7$~Mpc;][]{jordan2005} and a large estimated mass of the central
black hole~\citep[$M_{\rm BH} \simeq (3-6) \times
10^9~M_{\odot}$;][]{macchetto1997, gebhardt2009, walsh2013}\footnote{In this
Letter we adopt $M_{\rm BH} = 6.0\times 10^9M_{\odot}$ along with \citet{hada2011,
hada2012}.}, 1 milliarcsecond attains 0.08~pc or 140~Schwarzschild radii $(R_{\rm
s})$, providing an unprecedented opportunity to probe the jet formation processes
at its base.

The inner jet structure of M87 has been intensively investigated with
Very-Long-Baseline-Interferometry (VLBI). \citet{junor1999} discovered a
broadening of the jet opening angle at $\sim$100~$R_{\rm s}$ from the radio core
with an edge-brightened structure, and this was later confirmed in several
works~\citep{ly2004, dodson2006, krichbaum2006, ly2007, kovalev2007}.  More
recently, \citet[][hereafter AN12]{asada2012} discovered a maintanance of a
parabolic structure between a few $100~R_{\rm s}$ and $10^5~R_{\rm s}$ from the
core with a transition into a conical streamline above a location of
$\sim10^6~R_{\rm s}$. These results provide compelling evidence that the
collimation region is beginning to be resolved near the base of this jet.

However, the jet structure within $\sim$100$~R_{\rm s}$ remains unclear. Probing
the innermost region is essential to directly test theoretical models of
relativistic jet formation. Because the radio core at the jet base corresponds to
an optically-thick surface of synchrotron emission~\citep{bk1979}, previous
studies were not possible to determine the location of the central engine
relative to the radio core, preventing them from estimating the exact collimation
profile in this region.

Recently, we have overcome this difficulty~\citep[][hereafter H11]{hada2011}. By
measuring the core position shift~\citep{konigl1981, lobanov1998} with
multi-frequency phase-referencing Very-Long-Baseline-Array (VLBA) observations, we
have constrained the location of the central engine of the M87 jet as
$\sim$20$~R_{\rm s}$ upstream of the 43-GHz core. This allows us to probe the
radial profile of the jet structure as a function of the distance from the central
engine. Moreover, the determination of the frequency dependence of the core
position $r_{\rm c}\propto \nu^{-0.94\pm0.09}$ enables us to reveal the
structure of the innermost jet by investigating the multi-frequency properties of
the core. Indeed, the recent VLBI observation of M87 at 230~GHz detected a compact
core which is comparable to the size of event horizon~\citep{doeleman2012}, being
consistent with the core of this jet to be located in the black hole vicinity.

In this paper, we explore the collimation profile of the inner jet of M87,
especially based on the multi-frequency properties of the VLBI core as well as the
edge-brightened jet. The data analysis is described in the next section. In
section 3, we show our new results. In the final section, we discuss the derived
inner jet structure.

\begin{table*}[htbp]
 \begin{minipage}[t]{1.0\textwidth}
 \centering
  \caption{Multi-frequency VLBA observations of M87}
  \scalebox{1.00}{
  \begin{tabular}{lllccccccc}
   \hline
   \hline
   Freq. & Date & Beam $\theta_{\rm bm}$ & $I_{\rm rms}$ & \multicolumn{ 6}{c}{JMFIT Gaussian}       \\\cline{5-10}
   band  & & size, P.A. &  &  & $\theta_{\rm maj}$ & $\theta_{\rm min}$ & P.A. &
   $\theta_{20^{\circ}} (\equiv W_{\rm c})$ & $\frac{\theta_{20^{\circ}}}{\theta_{\rm bm, 20^{\circ}}}$ \\
   (GHz) &             & (mas)$^2$, (deg.)       & $\left(\frac{\rm mJy}{\rm beam}\right)$ &     & (mas) & (mas) & (deg.) & (mas) &  \\
   &             &  (a)                      & (b)  &      & (c)      & (d)   & (e)         & (f)        & (g) \\
   \hline
   2.3 & 2010 Apr  8 & 6.01 $\times$ 2.92, $-3$  & 0.79 &      & $3.26\pm 0.08$ & $< 1.06$ & $286\pm 3$ &  $<1.06$ & $<0.20$ \\
       & 2010 Apr 18 & 5.97 $\times$ 3.00, $-4$  & 0.66 &      & $3.47\pm 0.08$ & $< 1.06$ & $287\pm 3$ &  $<1.06$ & $<0.20$ \\\cline{5-10}
       &             &                           &      & Ave. & 3.37 & $< 1.06$ & 287 & $<1.06$ & $<0.20$ \\
       &             &                           &      &      & & & & & \\
   5.0 & 2010 Apr 8  & 2.63 $\times$ 1.36, 1     & 0.43 &      & $1.36\pm 0.06$ & $0.57\pm 0.16$ & $285\pm 4$ & $0.58\pm 0.17$ & 0.25\\
       & 2010 Apr 18 & 2.70 $\times$ 1.31, 0     & 0.37 &      & $1.38\pm 0.06$ & $0.56\pm 0.13$ & $288\pm 3$ & $0.56\pm 0.14$ & 0.24 \\\cline{5-10}
       &             &                           &      & Ave. & 1.37 & 0.57 & 287 & 0.57 & 0.25 \\
       &             &                           &      &      & & & & &  \\
   8.4 & 2009 May 23 & 1.48 $\times$ 0.55, $-9$  & 0.21 &      & $0.61\pm 0.07$ & $0.38\pm 0.05$ & $292\pm 5$ & $0.38\pm 0.06$ & 0.40 \\
       & 2010 Apr  8 & 1.67 $\times$ 0.84, 2     & 0.37 &      & $0.84\pm 0.04$ & $0.42\pm 0.06$ & $295\pm 4$ & $0.42\pm 0.07$ & 0.29 \\
       & 2010 Apr 18 & 1.63 $\times$ 0.85, $-3$  & 0.34 &      & $0.87\pm 0.05$ & $0.40\pm 0.08$ & $295\pm 4$ & $0.40\pm 0.09$ & 0.29 \\\cline{5-10}
       &             &                           &      & Ave. & 0.77 & 0.40 & 294 & 0.40 & 0.33 \\
       &             &                           &      &      & & & & & \\
   15  & 2009 Jan 7  & 1.04 $\times$ 0.47, $-3$  & 0.82 &      & $0.45\pm 0.02$ & $0.28\pm 0.01$ & $298\pm 5$ & $0.28\pm 0.02$ & 0.34 \\
       & 2009 Jul 5  & 1.00 $\times$ 0.46, $-8$  & 0.63 &      & $0.39\pm 0.05$ & $0.26\pm 0.03$ & $305\pm 7$ & $0.26\pm 0.06$ & 0.35 \\
       & 2010 Feb 11 & 1.00 $\times$ 0.43, $-2$  & 1.07 &      & $0.41\pm 0.02$ & $0.21\pm 0.04$ & $286\pm 2$ & $0.21\pm 0.04$ & 0.27 \\
       & 2010 Apr 8  & 0.94 $\times$ 0.48, $-$5  & 0.59 &      & $0.47\pm 0.02$ & $0.25\pm 0.04$ & $300\pm 5$ & $0.25\pm 0.04$ & 0.33 \\
       & 2010 Apr 18 & 0.92 $\times$ 0.49, $-10$ & 0.44 &      & $0.49\pm 0.02$ & $0.24\pm 0.02$ & $298\pm 3$ & $0.24\pm 0.03$ & 0.33 \\
       & 2010 Sep 29 & 1.00 $\times$ 0.45, $-2$  & 0.72 &      & $0.42\pm 0.07$ & $0.26\pm 0.03$ & $307\pm 4$ & $0.27\pm 0.08$ & 0.35 \\
       & 2011 May 21 & 0.93 $\times$ 0.43, $-5$  & 0.74 &      & $0.51\pm 0.05$ & $0.28\pm 0.05$ & $302\pm 5$ & $0.29\pm 0.07$ & 0.38 \\\cline{5-10}
       &             &                           &      & Ave. & 0.45 & 0.25 & 299 & 0.26 & 0.34 \\
       &             &                           &      &      & & & & & \\
  23.8 & 2010 Jan 18 & 0.60 $\times$ 0.29, $-8$  & 0.76 &      & $0.25\pm 0.04$ & $0.20\pm 0.02$ & $300\pm 3$ & $0.20\pm 0.04$ & 0.42 \\
       & 2010 Apr 4  & 0.63 $\times$ 0.29, $-4$  & 0.68 &      & $0.27\pm 0.02$ & $0.18\pm 0.03$ & $313\pm 7$ & $0.19\pm 0.04$ & 0.38 \\
       & 2010 Apr 8  & 0.54 $\times$ 0.29, $-5$  & 1.36 &      & $0.24\pm 0.03$ & $0.18\pm 0.04$ & $326\pm 6$ & $0.20\pm 0.05$ & 0.43 \\
       & 2010 Apr 18 & 0.57 $\times$ 0.28, $-13$ & 1.12 &      & $0.25\pm 0.02$ & $0.20\pm 0.03$ & $327\pm 3$ & $0.21\pm 0.04$ & 0.54 \\
       & 2010 May 1  & 0.62 $\times$ 0.28, $-9$  & 0.73 &      & $0.29\pm 0.02$ & $0.19\pm 0.03$ & $315\pm 6$ & $0.20\pm 0.04$ & 0.42 \\
       & 2010 May 15 & 0.64 $\times$ 0.29, $-12$ & 0.80 &      & $0.31\pm 0.03$ & $0.20\pm 0.04$ & $317\pm 4$ & $0.21\pm 0.05$ & 0.48 \\
       & 2010 May 30 & 0.62 $\times$ 0.28, $-11$ & 0.83 &      & $0.28\pm 0.02$ & $0.16\pm 0.05$ & $314\pm 5$ & $0.17\pm 0.05$ & 0.38 \\\cline{5-10}
       &             &                           &      & Ave. & 0.27 & 0.19 & 316 & 0.20 & 0.44 \\
       &             &                           &      &      & & & & &   \\
   43  & 2009 Mar 13 & 0.29 $\times$ 0.13, $-6$  & 0.55 &      & $0.15\pm 0.01$ & $0.13\pm 0.02$ & $313\pm 9$ & $0.13\pm 0.02$ & 0.61 \\
       & 2010 Jan 18 & 0.29 $\times$ 0.13, 2     & 1.01 &      & $0.11\pm 0.02$ & $0.10\pm 0.02$ & $262\pm 3$ & $0.10\pm 0.03$ & 0.39 \\
       & 2010 Apr 8  & 0.26 $\times$ 0.13, $-3$  & 0.91 &      & $0.13\pm 0.02$ & $0.11\pm 0.02$ & $354\pm 4$ & $0.13\pm 0.03$ & 0.57 \\
       & 2010 Apr 18 & 0.26 $\times$ 0.13, $-8$  & 1.12 &      & $0.13\pm 0.01$ & $0.11\pm 0.01$ & $358\pm 3$ & $0.13\pm 0.01$ & 0.63 \\
       & 2010 May 1  & 0.29 $\times$ 0.14, $-6$  & 0.89 &      & $0.11\pm 0.02$ & $0.11\pm 0.01$ & $318\pm 19$& $0.11\pm 0.02$ & 0.49 \\
       & 2010 May 15 & 0.29 $\times$ 0.14, $-9$  & 0.95 &      & $0.12\pm 0.01$ & $0.10\pm 0.02$ & $254\pm 6$ & $0.11\pm 0.02$ & 0.48 \\
       & 2010 May 30 & 0.30 $\times$ 0.15, $-6$  & 1.07 &      & $0.13\pm 0.01$ & $0.09\pm 0.01$ & $257\pm 8$ & $0.10\pm 0.02$ & 0.42 \\\cline{5-10}
       &             &                           &      & Ave. & 0.13 & 0.11 & 280 & 0.11 & 0.51 \\
       &             &                           &      &      & & & & &  \\
   86  & 2007 Feb 18 & 0.25 $\times$ 0.08, $-18$ & 6.47 &      & $0.079\pm 0.021$ & $0.065\pm 0.023$ & $52\pm 23$ & $0.074\pm 0.032$ & 0.64 \\
   \hline
  \end{tabular}
  }
 \medskip
 \end{minipage}
 \label{tab:tab1} Note: (a) uniformly-weighted beam size; (b) image noise level;
 (c),(d),(e) FWHM sizes of major/minor axes and position angle of derived model;
 (f) projected FWHM size of the model in $\rm{P.A.}=20^{\circ}$; (g) ratio of the
 Gaussian size devided by the beam size in $\rm{P.A.}=20^{\circ}$.
\end{table*}

   \begin{figure}[htbp]
   \centering
   \includegraphics[angle=0,width=1.0\columnwidth]{fig01.eps}
   \caption{Uniformly-weighted, averaged VLBA image of M87 at 43~GHz. Contours
   start from 3$\sigma$ image rms level and increasing by factors of 1.4.}
   \label{fig:image}
   \end{figure}

\section{Observations and Data Reduction}
We observed M87 with VLBA at 2, 5, 8.4, 15, 23.8 and 43~GHz on 2010 April 8 and
18. These are the same data presented in H11, where we investigated the core shift
of M87 using the phase-referencing technique relative to the nearby radio source
M84. Details of the observations and the data reductions processes are described
in H11.

To better constrain the averaged multi-frequency properties of the core and the
inner jet, we also analyzed VLBA archival data at 8.4, 15, 23.8 and 43~GHz. We
selected data observed after 2009 with sufficiently good qualities (all 10
stations participated, good $uv$-coverages, and thus high angular resolutions
obtained). Initial data calibration before imaging was performed using the
National Radio Astronomy Observatory (NRAO) Astronomical Image Processing System
(AIPS) based on the standard VLBI data reduction procedures. These data were not
acquired with phase-referencing mode.

Moreover, we added one VLBA archival data set at 86~GHz, which allows us to probe
the inner jet even closer to the black hole because of its higher transparency and
resolution. While several VLBA observations of M87 have been performed at 86~GHz,
we selected the data observed in 2007 February, because this is the only 86~GHz
data at present for which a reliaiable self-calibrated image has been published
with VLBA alone (peak-to-noise ratio of the self-calibrated image higher than
$\sim$70), as shown in \citet{rioja2011}. The observation is operated at 8
stations without Saint-Croix and Hancock. We analyzed this data based on the
procedures described in their paper.

Images were created in DIFMAP software with iterative phase/amplitude
self-calibration. We used uniform weighting scheme to produce high resolution
images.

\section{Results}
The M87 jet was clearly detected for all of the analyzed data. In Figure~1 we show
a representative image of M87 at 43~GHz, which was made by stacking the seven sets
of data. We confirmed the jet to be characterized by the compact core with the
edge-brightened structure directing an overall position angle of
P.A.$\sim$290$^{\circ}$.

\subsection{Model fitting on the core region}

In the present study, we aim at measuring the width of the innermost jet. For this
purpose, we fitted a single elliptical Gaussian model to each image with the AIPS
task JMFIT, and derived deconvolved parameters of the core region. Note that the
derived Gaussian size in this simple modeling could yield a larger size than that
of the true core (i.e., optical depth $\sim1$ surface) in the jet propagation
direction, because of blending of the optically-thin jet part. Also in the
transverse direction to the jet axis, this method would give a total size of the
true core plus surrounding emission, if the core region has a sub-structure that
is not resolved by VLBA baselines in this direction~\citep{dodson2006}. However,
here we are interested in measuring such a entire width of the innermost jet.

The results are summarized in Table~\ref{tab:tab1}. Most of the derived values
(especially for minor axes) are smaller than the beam size for each
data. Nonetheless, it is known that such sizes are measurable when the emitting
region is sufficeintly bright and robust self-calibration using as many as 10 VLBA
stations (8 at 86~GHz) can calibrate the fringe visibilities accurately. For the
M87 core, the derived sizes along the minor axes at 5, 8.4, 15, 23.8, 43 and
86~GHz correspond to amplitude decreases of 15\%, 18\%, 25\%, 30\%, 33\% and 30\%
at $\sim$60\% of the longest baseline (80M$\lambda$, 140M$\lambda$, 240M$\lambda$,
380M$\lambda$, 700M$\lambda$ and 1100M$\lambda$ respectively, which yield
effective angular resolutions in ${\rm P.A.=20^{\circ}}$). These decreases are
sufficiently larger than a typical VLBA amplitude calibration accuracy of
$\sim$5\% \citep[analogous discussion is presented in][]{kellermann1998}. At
86~GHz, previous Global-Millimeter-VLBI-Array observations report similar sizes to
the present value in Table~\ref{tab:tab1} \citep[$\lesssim 50$~$\mu$as or
$99\pm21$~$\mu$as;][respectively]{krichbaum2006, lee2008}.

We estimated the standard errors of the derived Gaussian parameters for each data
as follows. Formally, statistical size uncertainties purely based on
SNR~\citep[size of the fitted model devided by its peak-to-noise ratio;
e.g.,][]{fomalont1999} results in very small values for M87 (a level of a few
$\mu$as or smaller) because the core is bright at each frequency (peak-to-noise
$>70\sim1000$). However in practice, the model parameters are more strongly
affected by imperfect CLEAN/deconvolution processes under limited
$uv$-coverages. Then, we devided each individual data set into three subsets with
equal integration time, and repeated deconvolution and JMFIT processes for each
data individually. Through this procedure, we obtained three sets of
quasi-independent fitting results for each epoch, and the rms scatters can be
calculated for each parameter (i.e., major/minor axes and P.A.). These scatters
are adopted as realistic errors for each model parameter in Table~\ref{tab:tab1}.

As an additional check, we conducted a fake visibility analysis to examine how
precisely sizes smaller than beam sizes can be recovered~\citep[a similar analysis
is described in][]{taylor2004}; using the AIPS task UVCON, we created fake
visibilities which are equivalent to the derived Gaussian parameters at each
frequency with similar $uv$-coverages of the actual observations. We produced 10
independent visibility data sets at each frequency by adding random noises at a
level seen in the actual observations, repeated CLEAN/deconvolution and JMFIT on
each data, and calculated rms scatters of the recovered model parameters at each
frequency. We confirmed that these scatters were smaller than one-third of the
quoted errors in Table~\ref{tab:tab1} at frequencies smaller than 43~GHz, and less
than one-half at 86~GHz.

We note that, only regarding the minor axis of the 2-GHz core, JMFIT derived a
quite small size (less than 15\% of the beam size), which indicates a marginal
constraint.  So in Table~\ref{tab:tab1} we instead set one-fifth of the beam size
as an upper limit, which corresponds to $\sim$5\% amplitude decrease at the
longest baseline in this direction.

   \begin{figure}[bbb]
   \centering \includegraphics[angle=0,width=1.0\columnwidth]{fig02.eps}
   \caption{Gaussian sizes for the core region as a function of frequency. Sets of
   filled/open rectangle indicates projected sizes in ${\rm
   P.A.}=20^{\circ}/290^{\circ}$. Two lines are power-law fit to each distribution
   using 8.4, 15, 23.8 and 43~GHz data. A gray rectangle indicates the derived
   core size at 230~GHz~\citep{doeleman2012}. } \label{fig:coresize}
   \end{figure}

   \begin{figure*}[htbp]
   \centering \includegraphics[angle=0,width=0.75\textwidth]{fig03.eps}
   \caption{Jet width profile of M87 as a function of distance. A jet viewing
   angle of $i=15^{\circ}$ and a black hole mass of $M_{\rm
   BH}=6.0\times10^9M_{\odot}$ are adopted. Densely-sampled part indicates the
   width profile of the edge-brightened jet measured at multiple frequency with
   errors. Discretely-sampled part indicates the FWHM size profile for the core
   region projected in P.A=20$^{\circ}$ ($W_{\rm c}$). The core positions and the
   errors are estimated based on the previous astrometry
   result~\citep{hada2011}. A solid line indicates the best fit solution for the
   edge-brightened jet width $W_{\rm j}\propto r^{0.56}$, while a dashed line
   indicates $r^{0.76}$. The grey/black rectangles at the bottom left corner of
   the figure represent surfaces (event horizons) for
   non-spinning/maximally-spinning black holes, respectively.}
   \label{fig:jetwidth}
   \end{figure*}

In Figure~\ref{fig:coresize}, we show a Gaussian size projected along ${\rm
P.A.=20^{\circ}}$ as a function of frequency (hereafter we denote $W_{\rm
c}$)\footnote{Here $W_{\rm c}$ is defined as a FWHM of the derived ellipical
Gaussians when sliced along a position angle of $20^{\circ}$ i.e., $W_{\rm c}
\equiv \left[\frac{\theta_{\rm maj}^2 \theta_{\rm min}^2 (1 + \tan^2 (20^{\circ} -
{\rm P.A.})}{\theta_{\rm min}^2 + \theta_{\rm maj}^2 \tan^2 (20^{\circ} - {\rm
P.A.})}\right]^{1/2}$, where $\theta_{\rm maj}, \theta_{\rm min}$ and
P.A. indicate values in columns (c), (d) and (e) of Table~1. The errors in this
direction, shown in column (g) of Table~\ref{tab:tab1}, are calculated from usual
error propagation analysis using the errors $\theta_{\rm maj}, \theta_{\rm min}$
and P.A..}. This projection is perpendicular to an overall jet axis, correnponding
to a direction of the jet width. We found $W_{\rm c}$ to be clearly frequency
dependent, becoming smaller as frequency increases. To determine an averaged
frequency dependence of $W_{\rm c}$, we fitted a power law function to this plot
using the data at 5, 8.4, 15, 23.8, 43 and 86~GHz (2~GHz data are excluded because
of upper limits). We found the best-fit solution to be $W_{\rm c}(\nu) \propto
\nu^{\xi}$ where $\xi = -0.71\pm0.05$. Interestingly, when this frequency dependence
is extended toward higher frequency, its extrapolated size appears to result in a
similar value to the measured size by the recent 230~GHz VLBI
experiment~\citep{doeleman2012}, which is determined with a circular Gaussian
fit. For reference, we also show a Gaussian size distribution projected along a
jet propagation direction, but this does not seem to fit the 230~GHz core size.

\subsection{Jet width measurements}

In Figure~\ref{fig:jetwidth}, we show the radial profile of the M87 jet width as a
function of (de-projected) distance along jet \citep[assuming a jet inclination
angle $i=15^{\circ}$;][]{biretta1999, perlman2011}. Here we investigated the jet
width profile in the following two procedures: (1) measurements of $W_{\rm j}(r)$;
for the region where the jet is clearly resolved into a two-humped shape at each
frequency, we made transverse slices of the jet every 0.01$\sim$0.5~mas distance
along the jet, and each of the two-humped structure was fitted by a
double-Gaussian function. We then defined the separation between the outer sides
of the half-maximum points of the two Gaussains as jet width. When the jet at a
frequency becomes single-humped toward the upstream region, we measured the jet
width using higher frequency images because the jet is clearly resolved again into
a two-humped shape at higher resolutions. By using the images between 2 and
43~GHz, such a measurement was repeated over a distance of $\sim$10$^{5}$~$R_{\rm
s}$ down to $\sim$100~$R_{\rm s}$ along the jet. This process is basically the
same as that used in AN12, but we exclude measurements for the single-humped
region. We aligned radial positions of $W_{\rm j}(r)$ profiles between different
frequencies by adding the amount of core shift measured in H11 (described
below). This amount and the associated position error at each frequency provide
only tiny fractions of the distance where $W_{\rm j}(r)$ was measured at each
frequency (a level of $10^{-1\sim-2}$ at 43~GHz to $10^{-3}$ at 2~GHz), so
horizontal error bars for $W_{\rm j}(r)$ are removed in
Figure~\ref{fig:jetwidth}. At 86~GHz, we could not perform reliable measurements
of the jet width because the edge-brightened jet was only marginally imaged at a
level of (2--3)$\sigma$. (2) measurements of $W_{\rm c}(r)$; closer to the central
engine, we further constructed a radial distribution of $W_{\rm c}$. Because our
previous astrometric study H11 measured locations of the cores as 41, 69, 109,
188, 295 and 674~$\mu$as at 43, 23.8, 15, 8.4, 5 and 2~GHz from the convergence
point of the core shift (in R.A. direction or P.A$=270^{\circ}$), we can set their
de-projected distances along the jet (P.A.$=290^{\circ}$) as 24, 40, 63, 108, 170
and 388~$R_{\rm s}$ for $i=15^{\circ}$, respectively. For 86/230~GHz cores, we can
also set their de-projected positions as 12 and 5~$R_{\rm s}$ by assuming the same
asymptotic relation ($r_{\rm c}\propto\nu^{-0.94}$; H11) for the upstream of the
43~GHz core. Here we assume that the central engine is located at the convergence
point of the core shift specified in H11.

We confirmed that the edge-brightened region is well-expressed as a parabolic
structure of $W_{\rm j}(r)\propto r^{0.56\pm0.03}$ (solid line in Figure~3). This
is in good agreement with their finding $r^{0.58\pm0.02}$ in AN12. For the region
around $\sim$100~$R_{\rm s}$, where the independent measurements of $W_{\rm j}$
and $W_{\rm c}$ are overlapped each other, $W_{\rm c}$ at 5 and 8,4~GHz smoothly
connect with $W_{\rm j}$ at 43~GHz. The combination of the core size ($W_{\rm
c}\propto \nu^{\xi}$ where $\xi=-0.71\pm0.05$) and core position ($r\propto
\nu^{\alpha}$ where $\alpha={-0.94\pm0.09}$; H11) yields a radial dependence of
$W_{\rm c}$ as $W_{\rm c}(r) \propto r^{\frac{\xi}{\alpha}} = r^{0.76\pm0.13}$
(dotted line in Figure~3), which is slightly steeper than that of the outer jet
$W_{\rm j}(r)$, although the uncertainty is still large. In the present result, it
is still difficult to distinguish whether $W_{\rm c}$ at 5, 8.4, 15 and 23.8~GHz
are on the solid line or the dashed line due to their position uncertainties in
addition to those of sizes. On the other hand, $W_{\rm c}$ at 43 and 86~GHz tends
to be below the solid line. At 230~GHz, the exact profile cannot be discriminated
again because the data is totally dominated by its position uncertainty.

We note that the two methods used for $W_{\rm j}(r)$ (a double-Gaussain fit on
each slice image) and $W_{\rm c}(r)$ (based on a single two-dimensional elliptical
Gaussian model on each 2-D image) are different from each other. Nevertheless, the
observed consistencies of the values between the two methods were confirmed in the
overlapped region, indicating that $W_{\rm c}$ is actually a good tracer for the
width of the innermost jet region.

\section{Discussion}
Probing the collimation profile of the M87 jet is crucial to understand the
formation processes of relativistic jets.  Although the actual energetics of the
M87 jet is still under debate~\citep[e.g.,][]{abdo2009}, here we focus on the
framework of magnetohydrodynamic (MHD) jets, because it is widely explored as the
most successful scenario for jet production.

\subsection{$r\gtrsim100~R_{\rm s}$ Region}
Theoretically, the shape of a magnetized jet is determined by the detailed force
balance across the poloidal field lines. This is described as the trans-field
equation, which was originally derived by \citet{okamoto1975} for steady,
axisymmetric, rotating magnetized flow in a gravitational field without external
pressure. Later, it is invoked that magnetic hoop stresses associated with
toroidal field lines play a major role to realize a global collimation of a
magnetized jet~\citep[]{bp1982, heyvaerts1989, sakurai1985, chiueh1991}.

Here we observationally confirmed that the M87 jet is well characterized as a
parabolic collimation profile between $\sim$100 and $\sim$10$^{5}~R_{\rm s}$. This
is consistent with the prior work by AN12, where they also found a transition into
a conical shape above $\sim$$10^{6}~R_{\rm s}$. Regarding formation of a parabolic
shape, recent theoretical studies indicate the importance of external pressure
support at the jet boundary. \citet{okamoto1999} analytically showed that hoop
stresses alone would not realize global collimation, and numerical studies also
come along this line~\citep[e.g.,][]{nakamura2006, komissarov2007,
komissarov2009, toma2013}.
 \citet{komissarov2009} shows that when the external gas
pressure follows $p_{\rm ext}\propto r^{-a}$ where $a \lesssim 2$, the jet
maintains parabolic as $W_{\rm j}\propto r^{a/4}$, whereas for $a > 2$ the jet
eventually becomes conical due to insufficient external support. If the observed
radio emission of M87 traces the exterior edge of a magnetized jet, the measured
width profile suggests $a \sim2$. As a source of such confinement medium, AN12
propose an intersteller medium bounded by the gravitational influence of the
central black hole, such as a giant ADAF~\citep{narayan2011}. We add to note that
a purely hydrodynamic (HD) jet is also possible to produce the gradual parabolic
collimation of M87~\citep[e.g.,][]{bromberg2009}.

Interestingly, at the end of the collimation region, both AN12 and
\citet{bromberg2009} suggest HST-1 (a peculiar knot at a deprojected distance
$\gtrsim$120~pc or $2\times10^5R_{\rm s}$) as a stationary shocked feature
resulting from overcollimation of the inner jet, which is originally proposed by
\citet{stawarz2006} and \citet{cheung2007} to explain the broadband flaring
activity. While our recent kimematic observations of HST-1 shows clear evidence of
superluminal motions of the overall structure, a weak, slowly-moving feature is
also found in the upstream of HST-1~\citep{giroletti2012}, which indeed could be
related to a recollimation shock.

  \begin{figure}[ttt]
   \centering \includegraphics[angle=0,width=1.01\columnwidth]{fig04.eps}
   \caption{The same image as Fig.~1 but convolved with a circular Gaussian beam
   of 0.14~mas diameter, yielding roughly twice higher resolution in north-south
   direction. Contours in the inner region starts from 20$\sigma$ image rms and
   increasing by factors of 1.4. The length bars correspond to projected scale.}
   \label{fig:m87q_sr}
  \end{figure}

\subsection{$r\lesssim100~R_{\rm s}$ Region}

For the first time, we have revealed a detailed collimation profile down to $r\sim
10~R_{\rm s}$ by investigating the multi-frequency properties of the radio
core. An intriguing point here is that the measured collimation profile suggests a
possible tendency of a wider jet opening angle around $\sim$10 to $\sim$100~$R_{\rm
s}$ from the central engine.

Since the two methods of our width measurements are switched around
$r\sim100~R_{\rm s}$, one could speculate that the profile change is related to
some systematic effects due to the different methods. Then, to check the
two-humped jet shape further close to the core ($r\lesssim 100~R_{\rm s}$) more
clearly, we created a 43-GHz image convolved with a slightly higher resolution (a
circular beam of 0.14~mas diameter). The image is shown in Figure 4. Above
$\sim$0.4~mas downstream of the core (de-projected distance $\sim$220~$R_{\rm s}$
for $i=15^{\circ}$), where the jet is parabolic on logarithmic distance, the two
ridges are already oriented into a similar direction (dark-blue region in Figure
4). On the other hand, the opening angle made by the two ridges appear to broaden
more rapidly within $\sim$0.3~mas of the core (the region with contours in Figure
4), resulting in a more radially-oriented structure near the base. Such a tendency
is consistent with the observed possible transition from the solid line to the
(steeper) dashed line around $r\sim100~R_{\rm s}$ in Figure 3.

A transition of jet collimation profile near the black hole is actually suggested
from some of theoretical aspects. In the framework of relativistic MHD jet models,
most of the energy conversion from magnetic-to-kinetic occurs after the flow
passes through the fast-magnetosonic point~\citep[``magnetic nozzle''
effect;][]{li1992, vlahakis2003}. Beyond this point, the magnetized jet starts to
be collimated asymptotically into a parabolic shape, because the increasing plasma
inertia winds the field lines into toroidal directions and thus amplifies the hoop
stresses~\citep[e.g.,][]{tomimatsu2003}. The radius of the fast point is typically
a few times the light cylinder radius $R_{\rm lc}$~\citep{li1992}, where $R_{\rm
lc}$ is of the order of (1$\sim$5)~$R_{\rm s}$~\citep{komissarov2007}. Thus, if
the M87 jet is magnetically launched, the observed possible transition of the jet
shape around $10\sim100~R_{\rm s}$ could be explained by this process. Moreover,
the jet in this scale is likely to have complicated interactions with surrounding
medium such as accretion flow, corona and disk
wind~\citep[e.g.,][]{mckinney2006}. Their geometries and the local pressure
balance at the jet boundary would affect the initial jet shape. Alternatively,
such a change of the jet shape could happen as an apparent effect due to
projection, if the jet inclination angle is not constant down to the black
hole~\citep{mckinney2013}.

It is interesting to note that a time-averaged dependence $\nu^{-0.71}$ of $W_{\rm
c}$ appears to connect with the 230~GHz core size. However, this apparent
connection should be compared in a cautious manner; the $uv$-coverage used in
\citet{doeleman2012} yields the highest ($\sim$3000M$\lambda$) angular resolution
\textit{along} the jet direction for M87, while $\sim$5 times shorter projected
baselines transverse to the jet. Thus the derived size of $\sim$40~$\mu$as by
their circular Gaussian fit could be more weighted to the structure along the jet,
unless the brightness pattern of the jet base (when projected toward us) is
actually close to a circular shape. To clarify the exact relationship of the core
size at the higher frequency side, the addition of north-south baselines in the
future Event-Horizon-Telescope array is crucial, which can be realized by
including Chilean stations such as ASTE, APEX and ALMA~\citep{broderick2009,
doeleman2009}.

The results presented here have newly shed light on the crucial issues which
should be addressed more rigorously in future observations; where is the exact
location of the profile change between $\sim$10 and $\sim$100~$R_{\rm s}$ and how
does it change (e.g., a sharp break or a gradual transition)?  In addition,
simultaneous observations at multiple frequencies are important; a dynamical
timescale of our target region is an order of $t_{\rm dyn}\sim 10~R_{\rm s}/c \sim
7$~days, so the jet structure (size and position of the core) can be variable on
this timescale~\citep[e.g.,][]{acciari2009}. To address these issues, we are
currently conducting new high-sensitivity VLBA observations at 86~GHz in
combination with quasi-simultaneous sessions at lower frequencies, which allows
more robust investigations of the jet structure within $100~R_{\rm s}$ and thus
test some specific models more quantitatively. Finally, we also stress the
significance of future imaging opportunities with RadioAstron at 22~GHz or
global-VLBI at 43/86~GHz including ALMA baselines, because these will provide
images at drastically improved resolutions, especially in the direction transverse
to the M87 jet.

\acknowledgments

We acknowledge the anonymous referee for his/her careful review and suggestions
for improving the paper. We thank K.~Asada and M.~Nakamura for valuable
discussion. We are also grateful to M.~Takahashi, A.~Tomimatsu, M.~Rioja,
R.~Dodson, Y.~Asaki, S.~Mineshige, S.~Kameno, K.~Sekimoto, T.~Tatematsu and
M.~Inoue for useful comments. The Very-Long-Baseline-Array is operated by National
Radio Astronomy Observatory, a facility of the National Science Foundation,
operated under cooperative agreement by Associated Universities, Inc. This work
made use of the Swinburne University of Technology software
correlator~\citep{deller2011}, developed as part of the Australian Major National
Research Facilities Programme and operated under license. This research has made
use of data from the MOJAVE database that is maintained by the MOJAVE
team~\citep{lister2009}. This work was partially supported by KAKENHI (24340042
and 24540240). Part of this work was done with the contribution of the Italian
Ministry of Foreign Affairs and University and Research for the collaboration
project between Italy and Japan. KH is supported by Canon Foundation between April
2012 and March 2013, and by the Research Fellowship from the Japan Society for the
Promotion of Science (JSPS) from April 2013.


\section{Introduction}

In quantum systems with purely unitary evolutions, the total energy is a conserved quantity, since the corresponding Hamiltonian always commutes with itself. However, this argument does not apply to  systems with Hamiltonians which are self-adjoint but not Hermitian \cite{Kurcz_NJP} and to open quantum systems \cite{Bartana}. The spontaneous emission of a photon is always related to the loss of energy from its source \cite{Hegerfeldt93,Dalibard,Carmichael}. For example, in laser sideband cooling, a red-detuned laser field excites an electronic state of a strongly confined ion via the annihilation of a phonon from its motion. When followed by spontaneous photon emission, the phonon is permanently lost \cite{Wineland}. Contrarily to common believe, we show here that even un-excited and un-driven quantum systems might constantly leak energy into their environment. The origin of the predicted effect are non-zero decay rates and the counter-rotating terms in the interaction Hamiltonian which are usually neglected as part of the RWA.

Although the validity of this approximation has been questioned in the past \cite{agarwal,knight}, it is commonly used to describe quantum optical systems. An exception is Hegerfeldt \cite{hegerfeldt2}, who shows that the counter-rotating terms in the interaction between two atoms and the free radiation field can result in a small violation of Einstein's causality. Zheng {\em et al.}~\cite{Zubairy} also avoided the RWA and predicted corrections to the spontaneous decay rate of a single atom at very short times. Recently, Werlang {\em et al.}~\cite{Werlang} pointed out that it might be possible to obtain photons by simply placing an atom inside an optical cavity but no quantitative predictions have been made and no justification for the relevant master equation has been given. When avoiding the RWA, it has to be avoided everywhere, also in the system-bath interaction.

This paper contains a rigorous derivation of the quantum optical master equation for bosonic systems which uses only the Born and the dipole approximation. We apply our results to individual quantum systems (trapped atoms and optical cavities), and to a composite quantum system consisting of many atoms inside an optical resonator. It is shown that for sufficiently large numbers of atoms inside the cavity, the stationary state photon emission rate can be as large as typical detector dark count rates. Its parameter dependence could be verified experimentally using currently available atom-cavity systems \cite{Trupke,Reichel,Esslinger}. Our calculations confirm the relevance of the effects predicted in \cite{Werlang} when the atom-cavity interactions are collectively enhanced \cite{Holstein,Shah}. Similar energy concentrating effects might contribute significantly to the sudden heating in sonoluminescence experiments \cite{SL} and are responsible for temperature limits in cooling experiments \cite{review,cool}.

There are five sections in this paper. Section \ref{Model} contains a rigorous derivation of the master equation of a single bosonic quantum system beyond the validity range of the RWA. Section \ref{single} calculates the corresponding stationary photon emission rate. Assuming that the photon emission in the absence of external driving remains negligible in this case, we find that two of the constants in this master equation are approximately zero. Section \ref{comp} uses an analogously derived master equation to calculate the stationary state cavity photon emission rate for a composite quantum system consisting of many atoms inside an optical resonator. For feasible experimental parameters, we predict stationary state emission rates as high as $300$ photons per second. A detailed discussion of our results can be found in Section \ref{conc}.

\section{Master equation of a single quantum system beyond the RWA} \label{Model}

Let us begin by studying individual quantum systems, like optical cavities and tightly confined atoms, with the ability to emit photons. Their Hamiltonian $H$ in the Schr\"odinger picture and in the Born and the dipole approximation can be written as $H = H_0 + H_{\rm int}$ with \footnote{This form of $H_{\rm int}$ avoids fixing phase factors of states.}
\begin{eqnarray} \label{H}
H_0 &=& \hbar \omega ~s^+ s^- + \sum _{{\bf k},\lambda} \hbar \omega _k ~ a ^\dagger _{{\bf k} \lambda} a _{{\bf k} \lambda}~, \nonumber \\
H_{\rm int} &=& \sum_{{\bf k}, \lambda} \hbar ~ \big( g_{{\bf k} \lambda} ~ a _{{\bf k} \lambda} + \tilde g_{{\bf k} \lambda} ~ a _{{\bf k} \lambda}^\dagger \big) ~ s^+ + {\rm h.c.}
\end{eqnarray}
and $|\tilde g_{{\bf k} \lambda} | = |g_{{\bf k} \lambda}|$. The free radiation field consists of an infinite number of one-dimensional harmonic oscillators with wave vectors {\bf k}, frequencies $\omega_k$, polarisations $\lambda$, annihilation operators $a_{{\bf k} \lambda}$, and coupling constants $g _{{\bf k} \lambda}$ and $\tilde g _{{\bf k} \lambda}$. In case of an optical cavity, $\omega \equiv \omega_{\rm c}$ is the frequency of its field mode and the $s^\pm$ are the photon creation and annihilation boson operators $c^\dagger$ and $c$:
\begin{equation} \label{bose}
\left [ c, c^\dagger \right] =1 ~, ~~ \left [ c^\dagger, c^\dagger \right] = 0 = \left [ c, c \right] ~,
\end{equation}
In case of a large number of tightly confined two-level atoms with states $|0 \rangle $ and $| 1 \rangle $, $\hbar \omega \equiv \hbar \omega_0$ is the energy of the excited state $|1 \rangle$ of a single atom and the $s^\pm$ are the collective raising and lowering operators $S^\pm$ with
\begin{equation} \label{Dicke4}
\left [ S^-, S^+ \right ] = 1 ~, ~~ \left [ S^+, S^+ \right] = 0 = \left [ S^-, S^- \right] ~,
\end{equation}
as we shall see in the next paragraph.

Suppose the atoms are confined in a region with linear dimensions that are much smaller than the wavelength of the emitted light. Then $|{\bf k} \cdot ( {\bf r}_j - {\bf r}_i )| \ll 1$ for most particle positions ${\bf r}_i$ and ${\bf r}_j$ and for a wide range of wave vectors ${\bf k}$. This implies that all particles experience approximately the same $g _{{\bf k} \lambda}$ and $\tilde g _{{\bf k} \lambda}$. The Hamiltonian $H$ remains therefore the same, if we replace $s^+s^-$ in $H_0$ by $\sigma_3$ and $s^\pm$ in $H_{\rm int}$ by $\sigma^\pm$. Here $\sigma^\pm$ and $\sigma_3$ are defined as
\begin{eqnarray} \label{sigma}
\sigma^{\pm} \equiv \sum _{i=1} ^N \sigma _i ^{\pm} ~, ~~
\sigma _3 \equiv \sum_{i=1} ^N \sigma _{3i}
\end{eqnarray}
with $\sigma _{3i} = {1 \over 2} \left ( | 1 \rangle_{ii} \langle 1 |-| 0 \rangle_{ii} \langle 0 | \right )$, $\sigma_i^+ = | 1 \rangle_{ii} \langle 0 |$ and $\sigma_i ^- = | 0 \rangle_{ii} \langle 1 |$ being the su(2) spin-like operators of atom $i$:
\begin{equation} \label{su2}
\left [ \sigma _{3i} , \sigma_i ^{\pm} \right ] = \pm \sigma_i ^ {\pm} \, , ~ \left [ \sigma_i ^-  , \sigma_i ^+\right] = - 2 \sigma _{3i} ~ .
\end{equation}
If the atoms are initially all in their ground state, they evolve under the
action of the operators (\ref{sigma}) into the Dicke-symmetric states:
\begin{eqnarray}
| l \rangle _{\rm p} &\equiv& \left [ | 0_1 0_2 0_3 \dots 0_{N-l} 1_{N-l+1} 1_{N-l+2} \dots 1 \rangle + \dots \right . \nonumber\\
 && \left . + | 1_1 1_2 \dots 1_l 0_{l+1} 0_{l+2} \dots 0 \rangle \right ] / \left (
\begin{array}{c} N \\ l \end{array} \right )^{1/2}
\end{eqnarray}
which are the eigenstates of $\sigma_3$. The difference between excited and unexcited particles is counted by $\sigma _3$, since $_{\rm p} \langle l | \sigma _3 | l \rangle _{\rm p} = l - {1 \over 2} N$. For any $l$ we have \cite{cool}:
\begin{eqnarray}
\sigma ^+ ~ | l \rangle _{\rm p} &=& \sqrt{l+1} \sqrt{N-l}  ~ | l + 1 \rangle _{\rm p} ~ , \nonumber \\ \label{Dicke1}
\sigma ^- ~ | l \rangle _{\rm p} &=& \sqrt{N-(l-1)} \sqrt{l} ~ | l - 1 \rangle _{\rm p} ~ .
\end{eqnarray}
This shows that $\sigma ^{\pm}$ and $\sigma _3$ are represented on $| l
\rangle_{\rm p}$ by the Holstein-Primakoff non-linear boson realization
\cite{Holstein,Shah} $\sigma ^+ = \sqrt{N} S ^+ A_s$, $\sigma ^- = \sqrt{N}
A_s S^- $ with $ \sigma _3 = S^+ S^- - {1 \over 2} N$, $ A_s = \sqrt{1-S^+ S^-
  /N}$, $S^+ | l \rangle _{\rm p} = \sqrt{l+1} | l + 1 \rangle _{\rm p}$, and
$S^- | l \rangle _{\rm p} = \sqrt{l} | l - 1 \rangle _{\rm p}$ for any
$l$. The $\sigma$'s still satisfy the su(2) algebra (\ref{su2}). However, for
$N \gg l$, (\ref{Dicke1}) becomes
\begin{equation}
\sigma ^{\pm} ~ |l \rangle _{\rm p} = \sqrt{N} S^{\pm} ~ | l \rangle _{\rm p}
~.
\end{equation}
 Consequently, in the large $N$ limit, (\ref{su2}) contracts to the projective algebra e(2) \cite{Shah}
\begin{equation} \label{Dicke3}
\left[ S_3, S^{\pm} \right ] = \pm S^{\pm}  ~, ~~ \left [ S^-, S^+ \right ] = 1 ~,
\end{equation}
in terms of $S^{\pm}$ and $S_3 \equiv \sigma_3$. This means, the $s^\pm$ operators in the cavity case and in the many atom case are formally the same (cf.~(\ref{H})--(\ref{Dicke4})).

To derive the master equation of a single system, we assume that its state $|\varphi \rangle$ at $t=0$ is known. Moreover, we notice that spontaneously emitted photons leave at a very high speed and cannot be reabsorbed. The free radiation field is hence initially in a state with only a negligible photon population in the optical regime \cite{Hegerfeldt93,Dalibard,Carmichael}. Denoting this state by $|{\cal O} \rangle$, the (unnormalised) state vector of system and bath equals
\begin{equation} \label{kernel}
|{\cal O} \rangle |\varphi^0_{\rm I} \rangle = |{\cal O} \rangle \langle {\cal O}| ~ U_{\rm I} (\Delta t,0) ~ |{\cal O} \rangle |\varphi \rangle
\end{equation}
under the condition of {\em no} photon emission in $(0,\Delta t)$. In the interaction picture with respect to $H_0$, this equation can be calculated using second oder perturbation theory, even when $\Delta t \gg 1/\omega$. Doing so, we find
\begin{equation} \label{kernel2}
|\varphi^0_{\rm I} \rangle = \big[ 1 - A ~ s^+ s^- - B ~ s^- s^+ - C ~ s^{+2} - D ~ s^{-2} \big] ~ |\varphi \rangle
\end{equation}
with
\begin{eqnarray} \label{ABC}
A &=&  \int_0^{\Delta t} \!\! {\rm d}t \int_0^t \!\! {\rm d}t' ~ \sum _{{\bf k}, \lambda} g_{{\bf k} \lambda} \tilde g_{{\bf k} \lambda}^* ~ {\rm e}^{{\rm i}(\omega-\omega_k)(t-t')} ~, \nonumber \\
B &=& \int_0^{\Delta t} \!\! {\rm d}t \int_0^t \!\! {\rm d}t' ~ \sum _{{\bf k}, \lambda} g_{{\bf k} \lambda}^* \tilde g_{{\bf k} \lambda} ~ {\rm e}^{-{\rm i}(\omega+\omega_k)(t-t')} ~, \nonumber \\
C &=& \int_0^{\Delta t} \!\! {\rm d}t \int_0^t \!\! {\rm d}t' ~ \sum _{{\bf k}, \lambda} g_{{\bf k} \lambda} \tilde g_{{\bf k} \lambda} ~ {\rm e}^{{\rm i} (\omega -\omega_k) t + {\rm i} (\omega + \omega_k) t'} ~, \nonumber \\
D &=& \int_0^{\Delta t} \!\! {\rm d}t \int_0^t \!\! {\rm d}t' ~ \sum _{{\bf k}, \lambda} g_{{\bf k} \lambda}^* \tilde g_{{\bf k} \lambda}^* ~ {\rm e}^{- {\rm i} (\omega + \omega_k) t - {\rm i} (\omega - \omega_k) t'}  ~.~~~~~~
\end{eqnarray}
All four parameters could, in principle, be of first order in $\Delta t$ due to the sum over the infinitely many modes of the free radiation field.

In analogy to (\ref{kernel}), the (unnormalised) density matrix of the system in case of an emission equals
\begin{equation} \label{kernel4}
\rho^>_{\rm I} = {\rm Tr}_{\rm R} \left[ \sum _{{\bf k}, \lambda} a ^\dagger _{{\bf k} \lambda} a _{{\bf k} \lambda}
~ U_{\rm I} (\Delta t,0) ~ \tilde \rho ~ U_{\rm I}^\dagger (\Delta t,0)  \right]
\end{equation}
with $\tilde \rho = |{\cal O} \rangle \langle {\cal O}| \otimes \rho$ being the initial state of system and bath. Proceeding as above and using again second order perturbation theory, this yields
\begin{equation} \label{kernel5}
\rho^>_{\rm I} = \tilde A ~ s^- \rho s^+ + \tilde B ~ s^+ \rho s^- + \tilde C ~ s^- \rho s^- + \tilde D ~ s^+ \rho s^+ ~.
\end{equation}
The coefficients $\tilde A$, $\tilde B$, $\tilde C$, and $\tilde D$ are obtained when taking the complex conjugate of the coefficients $A$, $B$, $C$, and $D$ in Eq.~(\ref{ABC}) and extending the integration of the inner integral to $\Delta t$. To obtain relations between these coefficients, we decompose
\begin{equation}
\int_0^{\Delta t} \!\! {\rm d}t \int_0^{\Delta t} \!\! {\rm d}t' ... =
\int_0^{\Delta t} \!\! {\rm d}t \int_0^t \!\! {\rm d}t' ... + \int_0^{\Delta
  t} \!\! {\rm d}t \int_t^{\Delta t} \!\! {\rm d}t' ... ~.
\end{equation}
Substituting $u=\Delta t - t$ and $u' = \Delta t - t'$ in the second integral (which maps its area onto that of the first one) we find
\begin{eqnarray} \label{CD}
&& \tilde A = 2 {\rm Re} A ~, ~~  \tilde C = C^* + {\rm e}^{- 2 {\rm i} \omega \Delta t} ~ C ~,~~~ \nonumber \\
&& \tilde B = 2 {\rm Re} B ~, ~~ \tilde D = D^* + {\rm e}^{2 {\rm i} \omega \Delta t} ~ D ~.
\end{eqnarray}
Choosing the overall phase of $C$ accordingly \footnote{This is done by adjusting the phases of the states of the free radiation field which affects the $g_{{\bf k} \lambda}$ and $\tilde g_{{\bf k} \lambda}$ in (\ref{H}).}, the parameters $C$, $D$, $\tilde C$, and $\tilde D$ can hence be written as
\begin{equation}
C = D^* = {1 \over 2}f ~ \gamma_{\rm C} ~, ~~ \tilde C = \tilde D^* = f^* ~
\gamma_{\rm C} ~,
\end{equation}
with
\begin{equation}
f \equiv  {\rm e}^{{\rm i} \omega \Delta t} ~ \sin(\omega \Delta t) / \omega
\end{equation}
and with $\gamma_{\rm C}$ being a real but not specified function of $\Delta t$. One can easily check that this notation is consistent with $\tilde D = \tilde C^*$ (cf.~(\ref{ABC})) and with (\ref{CD}).

Averaging over the subensemble with and the subensemble without photon
emission (cf.~(\ref{kernel2}) and (\ref{kernel5})) at $\Delta t$ hence yields
the density matrix
\begin{eqnarray} \label{kernel7}
\rho_{\rm I} (\Delta t) &=& \rho - \big[ \big( A ~ s^+ s^- + B ~ s^- s^+) ~ \rho + {\rm h.c.} \big] \nonumber \\
&& \hspace*{-0.4cm} - {1 \over 2} \gamma_{\rm C} ~ \big[ \big( f ~ s^{+2} + {\rm h.c.} \big) ~ \rho + {\rm h.c.} \big] + 2 {\rm Re} A ~ s^- \rho s^+ \nonumber \\
&& \hspace*{-0.4cm} +  2 {\rm Re} B ~  s^+ \rho s^- + \gamma_{\rm C} ~ \big[  f ~ s^+ \rho s^+ + {\rm h.c.} \big] ~.
\end{eqnarray}
In the following we return into the Schr\"odinger picture considering a master equation \footnote{This equation is different from the one in
    G. S. Agarwal, {\em  Quantum Optics}, Springer Tracts of Modern Physics
    Vol. {\bf 70} (Spinger-Verlag, Berlin 1974).}
\begin{eqnarray} \label{deltarho2}
\dot \rho &=& - {{\rm i} \over \hbar} \left[ H_{\rm cond} \rho - \rho H_{\rm cond}^\dagger \right] + {\cal R}(\rho) ~ , \nonumber \\
{\cal R}(\rho) &=& \gamma_{\rm A} ~ s^- \rho s^+ + \gamma_{\rm B} ~ s^+ \rho s^- + \gamma_{\rm C} ~ \big( s^- \rho s^- + {\rm h.c.} \big) ~ , \nonumber \\
H_{\rm cond} &=&- {{\rm i} \over 2} \hbar \big[ \gamma_{\rm A} ~ s^+ s^- + \gamma_{\rm B} ~ s^- s^+ + \gamma_{\rm C} ~ \big( s^{+2} + {\rm h.c.} \big) \big] \nonumber \\
&& +  \hbar {\widetilde \omega} ~ s^+ s^-
\end{eqnarray}
with $\gamma_{\rm A} = 2 {\rm Re} A/\Delta t$, $\gamma_{\rm B} =  2 {\rm Re} B/\Delta t$, and with ${\widetilde \omega}$ being the shifted bare transition frequency. Checking the validity of (\ref{deltarho2}) can be done easily by returning into the interaction picture and integrating $\dot \rho_{\rm I} (t)$ from zero to $\Delta t$. The result is indeed (\ref{kernel7}).

Before continuing, we remark that the master equation (\ref{deltarho2}) has been derived using second order perturbation theory. This means, it applies to quantum optical systems with the ability to spontaneously emit photons. The assumption of a Markovian bath has been avoided. Instead, we assumed rapidly repeated (absorbing) measurements whether or not a photon has been emitted \cite{Hegerfeldt93,Dalibard,Carmichael}. These measurements constantly reset the free radiation field into $ |{\cal O} \rangle$ and make the system dynamics on the coarse grained time scale $\Delta t$ with $1/\omega \ll \Delta t \ll 1/\gamma$ automatically Markovian. Predictions based on this assumption have already been found in good agreement with actual experiments \cite{Toschek,Schoen}.

\section{Photon emission from a single quantum system} \label{single}

Let us continue by calculating the probability density $I_\gamma =  {\rm Tr} ({\cal R}(\rho))$ for a photon emission of a system prepared in $\rho$,
\begin{eqnarray} \label{kernel3}
I_\gamma &=& \left \langle \gamma_{\rm A} ~ s^+ s^- + \gamma_{\rm B} ~ s^- s^+ + \gamma_{\rm C} ~ \left( s^{+2} + s^{-2} \right) \right \rangle ~.~~~~
\end{eqnarray}
Using (\ref{bose}) or (\ref{Dicke4}), respectively, and considering the time evolution of the expectation values $\mu_1 \equiv \langle s^+ s^- \rangle$, $\xi_1 \equiv {\rm i} \langle s^{-2} - s^{\dagger +2} \rangle$, and $\xi_2 \equiv \langle s^{-2} + s^{+2} \rangle$, we obtain a closed set of rate equations,
\begin{eqnarray}
&& \dot \mu_1 = - (\gamma_{\rm A} - \gamma_{\rm B}) ~ \mu_1 + \gamma_{\rm B} ~, \nonumber \\
&& \dot \xi_1 = - (\gamma_{\rm A} - \gamma_{\rm B}) ~ \xi_1 + 2 \widetilde \omega ~ \xi_2 ~, \nonumber \\
&& \dot \xi_2 = - (\gamma_{\rm A} - \gamma_{\rm B}) ~ \xi_2 - 2 \tilde \omega ~ \xi_1 - 2 \gamma_{\rm C} ~.
\end{eqnarray}
Setting these derivatives equal to zero, we find that the stationary photon emission rate of a single bosonic system (like an optical cavity or many tightly trapped atoms) is
\begin{eqnarray}
I_\gamma = {2 \gamma_{\rm A} \gamma_{\rm B} \over \gamma_{\rm A} - \gamma_{\rm B}} - {2 \gamma_{\rm C}^2 (\gamma_{\rm A} - \gamma_{\rm B}) \over  4 \widetilde \omega^2 + (\gamma_{\rm A} - \gamma_{\rm B})^2} ~.
\end{eqnarray}
No photon emissions occur in the absence of external driving only when
\begin{eqnarray} \label{last}
\gamma_{\rm B} = \gamma_{\rm C} = 0 ~,
\end{eqnarray}
as it is assumed almost everywhere in the literature \cite{Hegerfeldt93,Dalibard,Carmichael,Werlang}. However, this assumption relies strongly on how the integrals in (\ref{ABC}) are evaluated and whether relations like $\tilde D(\omega) = \tilde C(-\omega)$ are taken into account or not.

\section{Photon emission from a composite quantum system} \label{comp}

Let us now have a look at a large number $N$ of tightly confined atoms inside an optical cavity. The energy of this composite system is the sum of the free energy of both subsystems, their interaction with the free radiation field, and the interaction between the atoms and the cavity field which changes (\ref{H}) into
\begin{eqnarray} \label{H2}
H_0 &=& \hbar ~ \omega_{\rm c} ~c^\dagger c + \hbar \omega_0 ~ S^+ S^-  + \sum _{{\bf k},\lambda} \hbar \omega _k ~ a ^\dagger _{{\bf k} \lambda} a _{{\bf k} \lambda} ~ , \nonumber \\
H_{\rm int} &=& \sum_{{\bf k}, \lambda} \hbar \big( g_{{\bf k} \lambda} ~ a _{{\bf k} \lambda} + \tilde g_{{\bf k} \lambda} ~ a _{{\bf k} \lambda}^\dagger \big) ~ c^\dagger + \sqrt{N} \hbar ~ \big( q_{{\bf k} \lambda} ~ a _{{\bf k} \lambda} \nonumber \\
&& \hspace*{-0.5cm} + \tilde q_{{\bf k} \lambda} ~ a _{{\bf k} \lambda}^\dagger \big) ~ S^+ + \sqrt{N} \hbar g_{\rm c} ~ \big( c+c^\dagger \big) ~ S^+ + {\rm h.c.} ~~~~
\end{eqnarray}
with $g_{\rm c}$, $g_{{\bf k} \lambda}$, $\tilde g_{{\bf k} \lambda}$,
$q_{{\bf k} \lambda}$ and $\tilde q_{{\bf k} \lambda}$ being coupling
constants. For simplicity, the cavity photon states should be chosen such that
$g_{\rm c}$ becomes real. Proceeding as in the single system case, assuming
the same properties of the bath, and returning into the Schr\"odinger picture
we receive again the master equation (\ref{deltarho2}) but with
\begin{eqnarray} \label{last2}
{\cal R}(\rho) &=& \kappa~ c \rho c^\dagger + N\Gamma ~ S^+ \rho S^- ~, \nonumber \\
H_{\rm cond} &=& \hbar \Big( \widetilde \omega_{\rm c} - {{\rm i} \over 2} \kappa \Big) ~ c^\dagger c + \hbar \Big( \widetilde \omega_0 - {{\rm i} \over 2} N \Gamma \Big) ~ S^+ S^- \nonumber \\
&& + \sqrt{N} \hbar g_{\rm c} ~ \big( c+ c^\dagger \big) \big( S^+ + S^- \big)  ~.
\end{eqnarray}
Here $\widetilde \omega_{\rm c}$ and $\widetilde \omega_0$ denote the bare atom and cavity frequencies, $\kappa$ is the cavity decay rate, and $\Gamma$ is the decay rate of the excited state of a single atom. The crucial difference to the usual Jaynes-Cummings model \cite{Knight} is the presence of the $c S^-$ and the $c^\dagger S^+$ term in (\ref{H2}) which vanish in the RWA. As we shall see below, these operators result in a non-zero stationary state population in excited states and the continuous emission of photons, even without external driving.

To calculate this rate, we take a conservative point of view and neglect $\gamma_{\rm B}$ and $\gamma_{\rm C}$ as suggested in (\ref{last}), since this assures that no emissions occur in the absence of external driving in the single system case. Using (\ref{bose}), (\ref{Dicke4}), (\ref{deltarho2}), and (\ref{last2}) we obtain again a closed set of the rate equations:
\begin{eqnarray}
&&\dot \mu _1 = \sqrt{N} g_{\rm c} \eta _1 - \kappa \mu _1 \, , ~~
\dot \mu _2 = \sqrt{N}  g_{\rm c} \eta _2 - N \Gamma \mu _2 ~ , \nonumber \\
&&\dot \eta _1 =  2 \sqrt{N} g_{\rm c} (1 + 2 \mu_2 + \xi_4) + \widetilde \omega_0 \eta _3 + \widetilde \omega_{\rm c} \eta _4 - {\textstyle {1 \over 2}} \zeta \eta_1 ~ , \nonumber \\
&&\dot \eta _2 = 2 \sqrt{N}  g_{\rm c} (1 + 2 \mu_1 + \xi_2) + \widetilde \omega _0 \eta _4 + \widetilde \omega _{\rm c} \eta _3 - {\textstyle {1 \over 2}} \zeta \eta_2 ~ , \nonumber \\
&&\dot \eta _3 = - 2 \sqrt{N}  g_{\rm c} (\xi_1 + \xi_3) - \widetilde \omega _0 \eta _1 - \widetilde \omega _{\rm c} \eta _2 - {\textstyle {1 \over 2}} \zeta \eta _3 ~ , \nonumber \\
&&\dot \eta _4 = - \widetilde \omega _0 \eta _2 - \widetilde \omega_{\rm c} \eta _1 - {\textstyle {1 \over 2}} \zeta \eta_4 ~, \nonumber \\
&&\dot \xi _1 = 2 \sqrt{N} g_{\rm c} \eta_4 + 2 \widetilde \omega_{\rm c} \xi_2 - \kappa \xi_1 ~, \nonumber \\
&&\dot \xi _2 = - 2 \sqrt{N} g_{\rm c} \eta_1 - 2 \widetilde \omega_{\rm c} \xi_1 - \kappa \xi_2 ~, \nonumber \\
&&\dot \xi _3 = 2 \sqrt{N} g_{\rm c} \eta_4 + 2 \widetilde \omega_0 \xi_4 - N \Gamma \xi_3 ~, \nonumber \\
&&\dot \xi _4 = - 2 \sqrt{N} g_{\rm c} \eta_2 - 2 \widetilde \omega_0 \xi_3 - N \Gamma \xi_4
\end{eqnarray}
with $\mu_1 \equiv \langle c^\dagger c \rangle$, $\mu_2 \equiv \langle S^+ S^- \rangle$, $\eta_{1,2} \equiv {\rm i} \langle (S ^- \pm S ^+) (c \mp c^\dagger) \rangle$, $\eta _{3,4} \equiv\langle (S ^- \mp S ^+) (c \mp c^\dagger )\rangle$, $\xi_1 \equiv {\rm i} \langle c^2 - c^{\dagger 2} \rangle$, $\xi_2 \equiv \langle c^2 + c^{\dagger 2} \rangle$, $\xi_3 \equiv {\rm i} \langle S^{-2} - S^{+2} \rangle$, $\xi_4 \equiv \langle S^{-2} + S^{+2} \rangle$, and $\zeta \equiv \kappa + N \Gamma$. Combined with (\ref{kernel3}), the stationary state of these equations yields  the cavity photon emission rate
\begin{eqnarray} \label{IN}
I_{\kappa} &=& {N \zeta \kappa g_{\rm c}^2 \left [ \, 8 \zeta g_{\rm c}^2 + \zeta^2 \Gamma + 4 \Gamma \left ( \widetilde \omega _0 - \widetilde \omega_{\rm c} \right )^2 \, \right] \over 16 \zeta^2 g_{\rm c}^2 \widetilde \omega_0 \widetilde \omega_{\rm c} + 2 \zeta^2 \kappa \Gamma \left ( \widetilde \omega _0 ^2 + \widetilde \omega_{\rm c}^2 \right ) + 4 \kappa \Gamma \left( \widetilde \omega _0^2 - \widetilde \omega_{\rm c}^2 \right)^2} \nonumber \\
\end{eqnarray}
which applies for $N \Gamma, \sqrt{N} g_{\rm c}, \kappa \ll \widetilde \omega_0,\widetilde \omega_{\rm c}$. For example, the parameters of the recent cavity experiment with $^{85}$Rb $[4]$ combined with $N=10^4$ are expected to result in an $I_\kappa$ as large as $I_\kappa =301~\rm{s}^{-1}$ which can be detected experimentally (cf.~Fig.~\ref{fig2}).

\begin{figure}[t]
\begin{minipage}{\columnwidth}
\begin{center}
{\includegraphics[scale=0.7]{fig.eps}}
\end{center}
\caption{The cavity photon emission rate $I_\kappa$ as a function of time for $N=10^4$ atoms inside the resonator obtained from a numerical solution of the respective rate equations. Here $\omega_{\rm c} = \omega_0 = 384.2 \cdot 10^{12} \, {\rm s}^{-1}$ ($D_2$ line), $g_{\rm c} = 6.1\cdot 10^8 \, {\rm s}^{-1}$, $\Gamma =  1.9 \cdot 10^7 \, {\rm s}^{-1}$, and $\kappa = 1.3 \cdot 10^{10} \, {\rm s}^{-1}$, as in Ref.~\cite{Trupke}. After being initially empty, the cavity becomes populated --- even in the absence of external driving.} \label{fig2}
\end{minipage}
\end{figure}

\section{Conclusions} \label{conc}

Let us now discuss the same topic in a more physical way. Among the reasons usually produced to justify the RWA, there is the one of the time scale, since the counter-rotating terms oscillate very rapidly so that their contribution to the dynamics remains in general negligible with respect to the resonating ones \cite{Grynberg}. Other authors apply the RWA in order to preserve quantum numbers and energy \cite{Milonni,Schleich}, since this approximation drops all the processes transferring energy between non-resonating modes. However, in Nature, there exist processes, where there is a redistribution of energy among different system degrees of freedom making possible some amounts of system self-organization. In particular, one could examine the possibility of concentrating the total energy of the system into a subset of degrees of freedom producing a decrease of its entropy, which in order to avoid a violation of the second law of thermodynamics, would compel the release of energy to the environment, thus keeping the free energy constant. This is possible, of course, only when the system is open.

In this paper, we consequently examined the situation occurring when the
counter-rotating terms are not dropped. The avoidance of the RWA and the
consequent more exact solution of the dynamics of the system under consideration gives rise to non-trivial consequences. As a matter of fact, the mechanism of concentration of energy on a subset of degrees of freedom could help the understanding of the hitherto rather mysterious processes of self-organization. The mathematical analysis we have done above shows indeed that in a quantum system a leakage of energy can occur among
different degrees of freedom. This leakage is not necessarily triggered by an external pump of energy, but
could be also triggered by the virtual photons coming from the quantum
vacuum as, e.g., it occurs in the Casimir effect or in the Lamb shift. From the
standpoint of the receiving system, the origin of the triggering energy is not important as far as
the balance between the variations of energy and entropy is satisfied so
to keep the free energy constant. In this respect, we recall that the ratio between these variations is just
the temperature, as required by the thermodynamic definition:
\begin{eqnarray}
k_{\rm B} T &=& {{\mathrm d}U \over {\mathrm d}S} ~.
\end{eqnarray}

The interplay between the microscopic quantum dynamics and the thermal properties certainly deserves further analysis, which, however, is out of the scope of the present paper. Here we are limiting ourselves to a specific physical picture, which however, does not exclude other physical scenarios, as, for example, the conversion of
energy from the thermal bath phonons to leaking photons (as it occurs in the laser
cooling mechanism). More
appealing could be the occurrence of a dynamics, where a system is able
to reach a state having
a lower energy jumping over a separating barrier with the help of
virtual photons coming from the vacuum. The possibility of realization
of these scenarios needs of course further studies. First indications
along the above lines can be found in the literature \cite{Kurcz_NJP,hegerfeldt2,Zubairy,Werlang}. In all these examples, the system dynamics is irreversible.

In summary, we derived the master equation for a single bosonic system (an optical cavity and a large number of tightly confined particles) without making any approximation other than the usual dipole and Born approximation. We find that the effect of the counter-rotating terms in the interaction between a quantum optical system and its free radiation field might be annihilated by environment-induced measurements whether or not a photon has been emitted. Assuming that this is the case, we then show that these measurements cannot suppress the interaction between a large number of atoms and an optical cavity. The result is the continuous leakage of photons through the resonator mirrors, even in the absence of external driving. For sufficiently many atoms, a relatively strong signal might be created. Its dependence on the system parameters can be verified experimentally using optical cavities like those described in \cite{Trupke,Reichel,Esslinger}. We recognize that in order to better understand the physical mechanisms responsible for the mathematical results presented in this paper, some more work is needed which is in our plans for future publications.

\vspace*{0.05cm}

\noindent {\em Acknowledgement.} A. B. acknowledges a James Ellis University Research Fellowship from the Royal Society and the GCHQ. This work was supported in part by the UK Research Council EPSRC, the EU Research and Training Network EMALI, University of Salerno, and INFN.