text:
\section{Introduction}
CMS is a general purpose detector at the Large Hadron Collider (LHC) at CERN. Its innermost tracking device is the pixel detector, which consists of three barrel layers and two endcap disks both up and downstream\cite{TDR}. Its basic building blocks are highly segmented silicon sensors with corresponding readout chips (ROCs).  Each sensor segment (pixel cell) is connected to a charge sensitive amplifier in the corresponding pixel unit cell (PUC) of the ROC via an indium bump of about $20\mu$m diameter.
In the first barrel layer, the active area is 4.3cm from the interaction point. At this short distance the density of produced particles will be as high as $4\cdot 10^7$sec$^{-1}$cm$^{-2}$ at the full LHC design luminosity of $\mathcal{L}=10^{34}$sec$^{-1}$cm$^{-2}$. Due to inclined tracks and Lorentz drift of the produced signal charge in the 4T magnetic field several pixels might be hit by a single track forming hit clusters. In order to improve position reconstruction for clusters through signal interpolation the amount of charge produced in each pixel is measured. The ROCs have to record position and charge for all hit pixels with a time resolution of 25ns, which is the time between two LHC bunch crossings. These information have to bo stored on-chip during the CMS first level trigger latency of 3.2$\mu$s after which they are either read out or discarded. The ROCs are read out serially via 40MHz analog links.\\
The first prototype of the readout chip has been developed in the radiation hard DMILL process \cite{DMILL,kurt}. In order to improve performance and yield and to reduce costs, the design has been migrated to a commercial process with much smaller feature size\cite{Wolfram}. The second iteration of the translated chip, PSI46V2, is going to be the production version and is described in this article. \\
In section two the architecture of PSI46V2 is described. Section three focusses on the analog readout scheme and in section four measurements of the performance are presented.

\section{Chip architecture}

The readout chip PSI46V2 has been fabricated in a commercial 5-metal layer 0.25$\mu$m process available through CERN. The design has been made radiation tolerant by following special layout rules as proposed in \cite{radhard}. The chip integrates 1.3 million transistors in an area of 7.9mm$\times$ 9.8mm.  It can be divided into 3 functional blocks: a control and supply block in the chip periphery, an array of pixel unit cells organized in double columns, and the double column peripheries which control readout and trigger validation within double columns. The total number of pixels is 80$\times$52 with a pixel size of $100\mu$m $\times 150 \mu$m.

\subsection{Control and supply block}
The chip periphery houses various control and supply circuits. It contains
\begin{itemize}
\item A serial programming interface. This is an I$^2$C-like protocol, modified to run at a speed of 40 MHz. To accommodate this high speed the possibility to read back configuration data had to be given up. Nevertheless, there is a limited readback possibility through the analog data stream as will be shown below. Two low voltage differential signal (LVDS) pairs are needed for clock and data lines.
\item A fast signal decoder. First level triggers and commands for reset and calibration signal injection are coded into a single LVDS signal. This signal has to be decoded and distributed over the chip.
\item 21 8-bit digital-to-analog converters (DACs), five 4-bit DACs and one 3-bit DAC to adjust offsets, gains, thresholds, supply voltages, timings, etc.
\item 2 control registers to set the trigger latency, readout speed (40 MHz/20MHz) and range for calibration pulses and to enable/disable the ROC.
\item A bandgap voltage reference and 6 voltage regulators. Three of them can be wire-bonded to external filter capacitors. This leads to a good immunity against power ripple and reduces chip-to-chip cross talk. The ROC needs two external power supplies, 1.5V for the analog section and 2.5V for digital part. It consumes a total of about 120 mW which corresponds to only $29\mu$W per pixel. The voltage regulators are programmable and hence the voltages can be set for each chip separately.
\item An analog event generator. This is a circuit that collects the pixel hit information from the double columns and generates the output data stream as described in section 3.
\item An on-chip temperature sensor for monitoring cooling performance.
\item A cluster multiplicity counter. This is a fast trigger signal that could be used by the CMS first level trigger or for self-triggering in the lab when no external trigger is available. Two thresholds can be set for this mechanism: one sets the minimal number of hits within a double column below which hits are considered as background and are ignored, whereas the other one tunes the number of hit double columns above which a trigger signal is issued. Note, that even below this threshold a signal proportional to the number of clusters is available.
\end{itemize}

\subsection{Pixel unit cell}
\begin{figure*}
\begin{center}
\includegraphics[angle=-90,width=15cm]{puc_nim.eps}
\end{center}
\caption{Schematic view of a pixel unit cell}
\label{fig:puc}
\end{figure*}
A sketch of the PUC is shown in figure \ref{fig:puc}. It can be divided into an analog part (top) and the digital logic (bottom). The thick lines on the right are bus lines running along the double column.\\
The signal from the sensor enters a two stage charge sensitive pre-amplifier/shaper system. Alternatively, calibration signals can be injected through either a 4.8fF injection capacitor connected directly to the amplifier input node, or via the sensor through the air gap between a top metal plate in the ROC and the sensor. The former is used for trimming the comparator threshold while the later can be used to test the bump bond connections\cite{andrei}. A globally programmable current source at the input node compensates for sensor leakage current. \\
Zero suppression is performed with a comparator. A global threshold can be programmed for the whole chip. In order to compensate for local transistor mismatches each pixel has a 4-bit DAC to trim the threshold. Furthermore, a mask bit allows to disable noisy pixels. Once the comparator is above threshold the shaper output signal is stored in a sample-and-hold circuit (see below). The double column periphery is notified immediately through a fast hard-wired column OR. The pixel becomes insensitive and waits for a column readout token. When it arrives the analog signal is sent to the periphery together with the pixel row address. The token is then passed on and the pixel resumes data taking. Thus, dead-time is short but depends on the hit rate. The time needed for a column scan to finish is $\leq$ 50ns + (50ns$\times$ number of hits). \\
The shaper output signal is sampled after a fixed time delay which is controlled by a global register (i.e. once per ROC). The signal within the sample-and-hold circuit follows the rising edge of the shaper output quite fast while it decays much slower. This makes the peak of the signal sufficiently flat and the circuit is not very sensitive to this time delay. Due to timewalk and device mismatch the exact peaking time depends somewhat on the pulse height and varies a bit from pixel to pixel. The relative error in the measured pulse height due to a non perfect sampling point has been measured and is shown in figure \ref{fig:error}. If the delay is optimised for low signal charges (3500 electrons in figure \ref{fig:error}), the relative error for large signals is only a few per cent.

\begin{figure}
\begin{center}
\includegraphics[width=6.5cm]{sample_hold.eps}
\end{center}
\caption{Relative error of peak detection across a full chip for 2 different pulse heights}
\label{fig:error}
\end{figure}
\noindent Downloading the full detector configuration at run start will take $\lesssim$ 0.5 minute \cite{danek}. If needed, pixels can be reprogrammed during LHC machine orbits reserved for sub-detector calibration, however only at a limited rate. Thus single event upset (SEU) in the trim and mask bit storage cells has to be considered. A simple trick has been adopted to protect storage cells from SEU. This is illustrated in figure \ref{fig:SEU}.
If node (a) collects a large enough amount of charge, produced by a nuclear reaction, the stored logic level can be changed. The presence of the capacitor aided by Miller effect increases the critical charge and thus reduces the probability of an upset.

\begin{figure}[b]
\begin{center}
\includegraphics[width=4cm]{seu.eps}
\end{center}
\caption{SEU protected storage cell}
\label{fig:SEU}
\end{figure}
\noindent The SEU cross section for both protected and unprotected storage cells has been measured at PSI for 300 MeV/c pions. The result is $\sigma=2.6\cdot 10^{-16}$cm$^2$ per storage cell for protected and $\sigma=2.4\cdot10^{-14}$cm$^2$ for unprotected cells. This is an improvement of 2 orders of magnitude. In PSI46V2 all storage cells in the PUCs are protected (4 trim bits and one mask bit). In case of an SEU the flipped bits have to be reprogrammed through control links. Each link serves several modules/pannels. In the worst case one link controls 4 modules plus 4 half-modules in each of the two innermost barrel layers, i.e. a total of 192 ROCs or 96 ROCs per layer. At high luminosity LHC operation the expected SEU rate for this control link is $<3\cdot 10^{-2}$Hz. This means that in about 8 hours 0.1\% of the pixels changed their threshold. The occupancy of each pixel can be monitored online and pixels that show significant changes will be reprogrammed.

\subsection{Double column periphery}
The double column periphery controls the transfer of hit information from the pixels to the storage buffers (column drain mechanism) and performs trigger verification. A schematic view of the logic is shown in figure \ref{fig:peri}.\\
\begin{figure*}[t]
\begin{center}
\includegraphics[width=16cm]{dcol_peri.eps}
\end{center}
\caption{Schematic view of the double column periphery}
\label{fig:peri}
\end{figure*}
When the asynchronous fast column OR signal arrives at the periphery three actions are performed:
\begin{enumerate}
\item[1. ] The present value of a bunch crossing counter (WBC) is latched into one of 12 time stamp buffer cells. This is needed later on for trigger validation.
\item[2. ] The hits for this bunch crossing are acknowledged by the periphery. This tells the pixels to associate any later hits with another column drain and its corresponding bunch crossing. There are one active and up to two pending column drains allowed. Hits of any further bunch crossing are lost.
\item[3. ] A column drain is initiated by sending out a readout token to the first pixel.\\
\end{enumerate}
A pixel without a hit belonging to the active column drain just bypasses the token. This has been measured to happen at a rate of about 3.3 GHz, i.e. the token arrives at the first hit pixel within less than 50ns or 2 clock cycles. Hit information is transferred in parallel into the data buffer, one pixel per two clock cycles. The data buffer has a depth of 32 units. Each unit consists of a marker bit one analog and nine digital storage cells for pulse height and row address respectively. The marker bit indicates the beginning of a new event and thus is used to synchronize entries in the time stamp and the data buffers. \\
The oldest entry in the time stamp buffer is permanently compared to a second bunch crossing counter (SBC) which is delayed with respect to the WBC by a programmable amount which corresponds to the first level trigger latency. In case of agreement the time stamp is deleted and the presence of the CMS first level trigger signal is checked. Either the corresponding data is discarded or the column is set into readout mode. In this mode, the double column stops data acquisition in order to prevent overwriting of valid data. It waits for the readout token to arrive and sends data to the chip periphery. Afterwards the double column resets itself and hence cannot have further valid hits for the duration of the trigger latency. Data loss also occurs when one of the buffers is completely filled up. In case of a full data buffer a double column gets reset. If the time stamp buffer is full, data acquisition is paused until the next buffer cell is freed.

\section{Readout scheme and chip readout format}
\begin{table}[b]
\small{
	\begin{tabular}{|l|c|c|c|}\hline
	Radius [cm] & \makebox[1cm]{4} & \makebox[1.2cm]{7} & \makebox[1.2cm]{11}\\ \hline
	pixel busy 	& 		0.21\% & 	0.078\% & 	0.044\% \\
	column drain busy & 		0.25\% & 	0.020\% & 	0.004\% \\
	time stamp buffer overflow & 	0.17\% & 	0.001\% & 	0 \\
	data buffer overflow & 		0.17\% & 	0.081\% & 	0.065\% \\ \hline
	1-buffer	&		1.0\% &		0.40\% &	0.26\% \\
	readout 	&		1.0\% &		0.20\%	&	0.16\%	\\
	double column reset & 		1.0\%  &	0.44\% &	0.26\% \\ \hline
	total &				3.8\% &		1.2\% &		0.79\% \\ \hline
	\end{tabular}
}
\label{table:loss}
\caption{Summary of simulated data loss for barrel modules at a luminosity of $10^{34}$sec$^{-1}$cm$^{-2}$ and 100kHz trigger rate. The upper part shows column drain losses, in the lower part readout related data losses are summarized}
\end{table}
A couple of design inherent data loss mechanisms have been mentioned so far. They are summarized in table \ref{table:loss} for barrel modules, a luminosity of $10^{34}\cdot$sec$^{-1}$cm$^{-2}$ and $100$ kHz first level trigger rate. A large contribution, labeled 'readout', comes from the fact that a double column is insensitive between the time it validates a trigger and the time it finishes the readout. This stresses the importance of having short readout times. Therefore an analog readout scheme running at 40 MHz (alternatively it can be switched to 20 MHz) has been adopted where the pixel address is coded into 6 discrete levels. Pixel address and signal pulse height can be transferred in 6 clock cycles. \\
In the endcap part of the detector several sensor tiles of different size are organized in so called blades which contain 24 ROCs on the front side and 21 ROCs on the back side. In the barrel, sensors connected to 16 (8) ROCs are called modules (half modules). The ROCs of each side of a blade or of a (half) module are read out in a daisy chained way. A token bit, controlled by a special token bit manager chip (TBM,\cite{ed}), is sent to the first ROC. The ROC sends its data to the TBM and passes the token bit to the next ROC, until the token bit gets back to the TBM. The TBM amplifies the signals from the ROCs, adds a header and a trailer and drives the signals out to the detector supply tube. \\
The readout of a single pixel hit is shown in figure \ref{fig:ro}. It starts with a header of 3 clock cycles. A large negative signal level well outside of the range of pixel data (ultra-black) followed by a zero differential level (black) separates the individual ROCs in the blade/module data stream. No other identification is sent. In order to get the ROC ID the DAQ unit has to count the chip headers. In the third clock cycle of the chip header a signal inversely proportional to the value of the last addressed DAC is sent. This is the only way information about the configuration of the ROCs can be read back. It follows the double column address (2 cycles), the row address (3 cycles) and the analog pulse height for each hit pixel.
\begin{figure}
\begin{center}
\includegraphics[width=7.5cm]{ro.eps}
\end{center}
\caption{Readout sequence of a ROC with one pixel hit}
\label{fig:ro}
\end{figure}
Figure \ref{fig:memory} shows the signal output of a barrel module as it arrives at the electro-optical converter. Shown are two different address level sequences. As can be seen, the risetime of the signals is short enough to allow for a tolerance of the sampling point in the DAQ unit of $\approx$ 6ns. The measured sampling distribution for a ROC is shown in figure \ref{fig:levels}. The address levels are clearly separated and allow a reliable reconstruction of the pixel addresses.
\begin{figure}
\begin{center}
\includegraphics[width=6cm]{memory.eps}
\end{center}
\caption{Readout signal output of a barrel module for two different address sequences}
\label{fig:memory}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=7.5cm]{addlevels.eps}
\end{center}
\caption{Histogram of the sampled address levels of a ROC. The levels are cleanly separated}
\label{fig:levels}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[width=7.5cm]{analog_out.eps}
\end{center}
\caption{Analog signal transmission.}
\label{fig:anout}
\end{figure}
\begin{figure*}
\begin{center}
\includegraphics[width=16cm]{Threshold.eps}
\end{center}
\caption{Threshold distribution of a ROC before (a) and after (b) trimming}
\label{fig:trim}
\end{figure*}

\section{Measurements}
In this section lab measurements of the performance of the ROC are shown.
Figure \ref{fig:anout} shows the pulse height response to calibration signal injection. It is slightly non-linear at low charges and saturates for high charges above about 2 MIPS.
Injection capacitances have been calibrated with X-ray sources. K$_\alpha$ lines of 6 different isotopes between 8.04keV and 44.2keV have been used. These measurements are also shown in figure \ref{fig:anout}.\\
It is important to have a threshold distribution as uniform as possible. On one hand the threshold must be set well above the amplifier noise in order not to flood the system with random hits. On the other hand the position resolution depends almost linearly on the threshold (see \cite{TDR}). The width of a gaussion fit to the untrimmed threshold distribution of a ROC is around 300 electrons as shown in figure \ref{fig:trim} (a). Adjusting the trim bits is one of the most important steps in the module characterization process \cite{andrei}.
\begin{figure}[h]
\begin{center}
\includegraphics[width=7.5cm]{timewalk.eps}
\end{center}
\caption{Timewalk for 3 different thresholds. The offsets of the 3 curves are arbitrary for better readability}
\label{fig:timewalk}
\end{figure}
After trimming, the threshold dispersion can be improve considerably to typically 80 electrons as shown in figure \ref{fig:trim} (b). The intrinsic noise of the preamp/shaper has been measured to be $\lesssim 190$ electrons \cite{andrei}.\\
The threshold described so far is not directly relevant. The time between a particle crossing the sensor and the comparator going above threshold depends on the created ionization charge. Therefore it might happen that for low pulses the hit signal is delayed to the next bunch crossing and hence is lost during trigger validation. This timewalk can be measured by delaying the injected calibration pulses with respect to the chip clock. The result of such scans is shown in figure \ref{fig:timewalk} for three different thresholds. For an absolute threshold of 2500 electrons it is shown that the effectively usable threshold (in-time threshold) is at about 3200 electrons, i.e. the time difference between a signal of 3200 electrons and very high charges above 100k electrons is below 25ns.  \\
Test structures containing the analog section of the PUC have been irradiated with gamma rays up to a total dose of 13.2 Mrad. This corresponds to the absorbed dose in 2 years at high luminosity LHC operation for the 7cm barrel layer or 1 year for the innermost layer. Figure \ref{fig:irrad} shows the outputs of the shaper and the sample-and-hold circuit in transparent mode. Curves (a) show the response for unirradiated structures. After irradiation, the analog signals become somewhat slower (b), but can be brought back close to the original waveform by reprogramming the feedback settings (c).\\

\begin{figure}[h]
\begin{center}
\includegraphics[width=7cm]{irrad_struct.eps}
\end{center}
\caption{Output of sample-and-hold circuit (top) and shaper output (bottom) before and after irradiation.}
\label{fig:irrad}
\end{figure}

\section{Summary}
The final version of the CMS pixel readout chip has been designed in a $0.25\mu$m process. It is fully functional and measurements of the performance have been presented. The threshold can be tuned to values below 2500 electrons with a uniformity of 80 electrons. Due to timewalk the in-time threshold is somewhat higher at 3200 electrons. The readout uses an analog scheme with address levels coded into 6 levels. This has been proven to work reliably at 40 MHz as address decoding was 100\% correct.


text:
\section{Introduction}
\label{intro}

Statistical thermal models~\cite{thermalmodels} have been very successful in characterizing the thermodynamical
properties of the hadronic medium created in heavy-ion collisions at the chemical freeze-out. By simply
measuring the total multiplicities of particles, the temperature $T_f$ and the baryochemical potential
$\mu_{B,f}$ can be extracted with a satisfactory statistical confidence. However, the assumption of a single
$T_f$ entails a tension between multistrange and nonstrange baryons in the thermal fits made to the data
of the ALICE collaboration~\cite{alice}. The $\Xi$ and $\Omega$ baryons seems to prefer a larger freeze-out temperature
(with a difference of $\sim$16 MeV) than the $T_f$ for protons.

On the other hand, the computation of generalized susceptibilities by lattice-QCD shows that
the inflection point of the 2nd-order susceptibility---indicator of the crossover transition---is located
for the strange sector at a temperature which is around 15 MeV larger than the one for its light
counterpart~\cite{rene}. Therefore, there exist indications of a flavor hierarchy in (at least) two different
observables: the QCD phase transition and the freeze-out temperatures~\cite{sandeep}. This issue have created
much interest at this SQM2017 conference, e.g.~\cite{sqm}.

In this contribution we provide qualitative evidences of such a hierarchy by presenting two separate studies. The QCD deconfinement temperature will be accessed by the melting temperature of
hadrons in the PNJL model of strong interactions. The freeze-out temperature will be represented by
the decoupling temperature of an expanding system of particles using a relativistic transport model.

\section{Melting temperature hierarchy in the PNJL model~\label{PNJL}}

The PNJL model is an effective realization of the low-energy dynamics of QCD~\cite{njl}. Thanks to its
Lagrangian formulation it can be easily extended to finite densities and temperatures. The dynamical
degrees of freedom are $N_f = 3$ massive quarks. Nevertheless, colorless states like mesons and baryons
can also be addressed in this model with the use of few-body techniques~\cite{benjamin}. Mesons are modeled as
quark-antiquark bound states which are stable in vacuum. At higher temperatures mesons can melt
into pairs. This happens at the so-called Mott (or melting) temperature, which is understood as the
effective deconfinement temperature (as the PNJL model lacks true confinement).

Similar considerations apply to bound states of 2 quarks or diquarks, and to bound states of a
diquark and a quark. The later are identified with physical baryons when they belong to a singlet
color representation. In Ref.~\cite{benjamin} we computed the masses and Mott temperatures of mesons and
baryons in several flavor/spin channels. A summary of the melting temperatures is shown in Table~\ref{tab:njl}.
In addition, the baryons masses versus temperature (up to the Mott temperature) are plotted in Fig.~\ref{fig-1}.

\begin{table}[ht]
\centering
\caption{\label{tab:njl} Mott (or melting) temperatures of several mesons and baryons calculated in the PNJL model~\cite{benjamin}.}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
\bf{Meson} & $\pi$ & $K$ & $\eta$ & $\eta'$ & $\rho$ & $K^*$ & $\omega$ & \textcolor{red} {$\phi$} \\
\hline
$T_{Mott}$ (MeV)& 282 & 286 & 245 & 0 & 253 & 266 & 253 & \textcolor{red} {382} \\
\hline
\hline
\bf{Baryon} & \textcolor{red} {$p$} & $\Lambda$ & $\Sigma$ & \textcolor{red} {$\Xi$} & $\Delta$ &  $\Sigma^*$ & $\Xi^*$ & \textcolor{red} {$\Omega$} \\
\hline
$T_{Mott}$ (MeV)& \textcolor{red} {254} & 269 & 195 & \textcolor{red} {287} & 223 & 231 & 239 & \textcolor{red} {288} \\
\hline
\end{tabular}
\end{table}

\begin{figure}[ht]
\centering
\includegraphics[width=5.8cm]{octet_Pmum.pdf}
\includegraphics[width=5.8cm]{decuplet_Pmum.pdf}
\caption{Masses of baryons belonging to the flavor octet (left panel) and decuplet (right panel) representations
as functions of the temperature until the melting (or Mott) temperature.}
\label{fig-1}
\end{figure}

Several conclusions can be extracted from Table~\ref{tab:njl} and Fig.~\ref{fig-1}: {\bf 1)} The effective deconfinement
temperature is not universal. {\bf 2)} The melting temperatures of $\Xi$ and $\Omega$ are larger (ca. 30 MeV) than
the one for the proton. This relative ordering might be responsible for the situation observed in the
statistical thermal fits [2]. {\bf 3)} Multistrange states generally have larger melting temperatures (except
the $\Xi^*$). In particular, the $T_{Mott}$ of the $\phi$ meson (strange-antistrange pair) is abnormally large.

These results point to a hierarchy in the deconfinement conditions, with a larger temperature for
the strange sector than the light one, as reflected in lattice-QCD results for the susceptibilities~\cite{rene}.

\section{Expanding system and its decoupling}

In a completely different approach we have focused on the freeze-out mechanism and its dependence
on the particle properties. For this case we have used a transport model to describe a system of
relativistic particles interacting via constant cross sections. SMASH (Simulating Many Accelerated
Strongly-interacting Hadrons) is a newly developed transport model that aims to describe the hadronic
stage of heavy-ion collisions with applications from GSI energies to LHC physics~\cite{smash}. We have
adapted SMASH to work with a background Friedmann-Robertson-Walker metric so that the system
exhibits a Hubble-like expansion in time. We use a spherical volume filled with a large number of
particles, initialized with an isotropic, homogeneous spatial distribution.

For a particular initial condition, we were able to reproduce the exact analytical solution of the
massless Boltzmann equation found in Ref.~\cite{bazow}. This represents a nontrivial check for the SMASH
code in a far-from-equilibrium scenario~\cite{tindall}.

For massive particles, we apply a Hubble expansion whose initial rate is much smaller than the
scattering rate. In this regime the system is able to reach and maintain equilibrium during the expan-
sion. When the expansion rate becomes larger than the scattering rate, the equilibrium state cannot be
maintained anymore, as the number of collisions is not sufficient to redistribute the momentum.

\begin{figure}[ht]
\centering
\includegraphics[width=4.5cm]{Dec_fp_t0-1_20.pdf}
\includegraphics[width=4.5cm]{Dec_fp_t5_20.pdf}
\includegraphics[width=4.5cm]{Dec_fp_t20_20.pdf}
\caption{Distribution function at different times of the Hubble expansion as a function of the particle momentum.
From left to right $t=0.1, 5$ and $20$ fm. In solid line we show the predicted equilibrium distribution for each time.}
\label{fig:dec1}
\end{figure}

From Fig.~\ref{fig:dec1} we can observe that SMASH perfectly reproduces the theoretical distribution function
(solid line) until decoupling happens soon after the middle panel. At $t=20$ fm the distribution from
SMASH does not match the equilibrium expectations. Taking into account the absence of collisions
and the redshift in momentum, the distribution after the freeze-out (decoupling) reads

\begin{equation}
f_{freeze-out} (t,k) = f_{equilibrium} \left( t_D,k \textcolor{red} {a(t)/a_D} \right) \propto \exp \left[ - \frac{ \sqrt{ \left( k \textcolor{red} { a(t) /a_D} \right)^2+m^2} }{\textcolor{blue} {T_D}} \right] \ ,
\end{equation}
where $t_D$ is the decoupling time, $\textcolor{red} {a(t)}$ is the scale factor, and $\textcolor{red} {a_D} , \textcolor{blue} {T_D}$ are the value of the scale factor and the temperature at the decoupling time, respectively.

The fit to this function provides the value of the decoupling time and temperature. In the example
of Fig.~\ref{fig:dec1} it gives $t_D = 5.3 \pm 0.6$ fm. Repeating this procedure for particles with different masses and
cross sections, we obtain the results in Fig.~\ref{fig:dec2} for the extracted freeze-out times.

The main conclusions of this study are: {\bf 1)} The more massive particles freeze out earlier (at higher
temperature). {\bf 2)} The larger the cross section is, the later the freeze-out appears (lower temperature).
Two caveats are in order. This scenario is closer to the kinetic freeze-out (the absence of interactions)
rather than the chemical freeze-out (relevant for statistical thermal fits). So far, inelastic processes are absent in our study. In addition, we have not explicitly introduced strangeness in the system.
However, one can notice that strange hadrons generally entail larger masses, and smaller cross sections
in comparison with their lighter counterparts. According to the results in Fig.~\ref{fig:dec2}, the combined effect
would result in an earlier freeze-out for the hadrons with strangeness (i.e. higher values of $T_f$).

\begin{figure}[t]
\centering
\includegraphics[width=5.5cm]{tD_mass.pdf}
\includegraphics[width=5.5cm]{tD_cross.pdf}
\caption{Decoupling time versus the mass of the particles with fixed cross section $\sigma=40$ mb (left panel), and
versus their total cross section with fixed mass $m=2$ GeV (right panel).}
\label{fig:dec2}
\end{figure}

\section{Conclusion}

We provide evidences of a flavor-nonuniversal deconfinement and freeze-out conditions that support
previous results in this directions~\cite{sqm,sandeep,rene}. On one hand, using the PNJL model we observe that multistrange hadrons melt at higher temperatures rather systematically. This points towards an earlier
confinement transition for the strange sector in heavy-ion collisions. On the other hand, the decoupling mechanism in an expanding spacetime provides direct indications that more massive and less
interacting particles freeze out earlier and at higher temperatures. This observation supports the higher
freeze-out temperature found for (multi)strange hadrons in statistical thermal fits~\cite{alice}.

 \vspace{2mm}

\begin{acknowledgement}
Work supported by DAAD (Germany), Helmholtz Young Investigator Group VH-NG-822 (Helmholtz Association and GSI), the HIC for FAIR within the framework of the LOEWE program (Hesse, Germany), and
Project No.FPA2013-43425-P (Spain), and Program TOGETHER from R\'egion Pays de la Loire and the I3-Hadron Physics (EU). We acknowledge LOEWE-Center for Scientific Computing for computing resources.
\end{acknowledgement}

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}

Since the discovery of polarized broad emission lines in the
archetypal Seyfert 2 galaxy NGC 1068 (Antonucci \& Miller 1985),
the AGN Unified Model (UM)
was tested over the entire electromagnetic spectrum.
While it has been realized that its original
formulation was too simple to explain the
large body of observational data, it still
provides a useful framework for AGN studies.

Hard X--ray observations of Seyfert galaxies
have shown that absorption by circumnuclear gas
and dust, presumably with a toroidal geometry, is almost ubiquitous
among optically classified Type 2 objects.
Deep and wide X--ray surveys, combined with pointed observations of nearby
bright AGN, allowed to probe a large interval
of luminosities, redshifts and column densities and thus to test
the AGN UM in the X--ray band.

A wide range of absorbing column densities is observed among Seyfert 2
and is required to fit the X--ray background ({\sc xrb}) spectrum (Gilli et al 2007; Treister
et al. 2009). Absorption is much less common at high X--ray luminosities
with a trend similar to that observed at optical and infrared wavelengths
(Simpson 2005, Maiolino et al. 2007).
There are also hints of an increasing fraction of obscured AGN
towards high redshifts (La Franca et al. 2005; Treister et al. 2006).
Besides providing additional tests to the UM, the above trends are most likely
closely related to the growth and evolution of Supermassive Black Holes (SMBHs).
According to current models (e.g. Hopkins et al. 2008)
all the galaxies undergo a phase of heavy, possibly Compton Thick
($N_H > 10^{24}$ cm$^{-2}$) obscuration, strong  accretion
and star formation. The census of obscured AGN may thus provide
useful insights on our understanding of a key phase of
SMBH and galaxy co-evolution.
Due to the lack of sensitive X--ray observations above 10 keV, Compton
Thick AGN can be efficiently detected and recognized as such only in the
local Universe.

In the following we present {\it Suzaku} observations of a small sample
of hard X--ray selected Seyfert 2. The goal is to characterize the
physics and geometry of the obscuring gas and, a first step, to
investigate their local space density. We also briefly discuss the
search for the most obscured sources in various X--ray surveys
and the implications for the study of their properties at high redshifts.

\section{Suzaku observations of nearby Seyfert 2 galaxies}

We have conceived a program with {\it Suzaku} (Mitsuda et al. 2007) to
observe five nearby, relatively X--ray bright ($> 10^{-11}$ \cgs)
AGN. The sources were selected from the {\sc integral/ibis}
(Beckmann et al. 2006) and {\sc swift/bat} (Markwardt et al. 2005) catalogues.
The column densities, as inferred from archival {\it Chandra}
and XMM--{\it Newton} observations, are of the order of $10^{23-24}$ cm$^{-2}$,
though affected by large errors.
For all the sources in the sample, a significant detection is achieved
with the hard X--ray detector up to 40--50 keV along with a good quality X--ray spectrum
with the {\sc xis ccd} below 10 keV (Fig.~1 and Comastri et al. 2009).
The high energy ($>$ 2-3 keV) spectra are fitted with an absorbed  power law plus
a reflaction component and an iron line.

\begin{figure}[t]
\centering
\psbox[xsize=8cm]
{comastri-andrea_fig_1.ps}
\caption{The Suzaku spectra of the Seyfert 2 galaxy sample. The open circles at
high energies are from Integral.}
\end{figure}

Not surprisingly, heavy absorption with column densities in excess
of $10^{23}$ cm$^{-2}$ is measured in all the sources. Three of them are
Compton Thick ($N_H > 10^{24}$ cm$^{-2}$) and among those, two are best
fitted by a reflection dominated spectrum.
A summary of the spectral
fit parameters, relevant for the present discussion, is reported in Table 1.
Soft X--ray emission, in excess of that expected by an extrapolation
to lower energy of the absorbed spectrum, is clearly observed in all
objects with only one exception.
The low energy spectrum is fitted with two different models: (i) a
partial covering (leaky absorber) model and (ii) a power law
plus narrow Gaussian lines to approximate the emission of circumnuclear
gas photoionized by the central nucleus.

\subsection{Partial covering model}

The fits with a partial covering model, which provides a good description
of the broad band spectra, allow us to compute the fraction ($f$) of
primary X--ray emission scattered into the line of sight and related to the
covering fraction  of the absorbing matter (e.g. the postulated torus
in AGN UM). Interesting enough, the measured values
of $f$ and the intensity of the reflection component ($R$) follow the correlation
suggested by Eguchi et al. (2009) based on the analysis of a small sample
of 6 Seyfert 2 galaxies detected by {\sc Swift/Bat} and selected in a very similar
way (Fig.~2).
Following Eguchi et al (2009), sources with a low scattering fraction, dubbed ``New Type"
AGN, are associated to a geometrically and optically thick configuration of the obscuring gas
seen rather face on. On the basis of a solid angle argument, they predict
a large population of heavily Compton Thick
AGN ($N_H > 10^{25}$ cm$^{-2}$) with extremely low scattering fraction
which would remain largely undetected even in hard X--ray observations.
While it may be premature to invoke the presence of a new population on the
basis of the present data, the existence of fully covered, heavily obscured AGN
would have important consequences for the census of SMBHs.

\begin{figure}[t]
\centering
\psbox[xsize=7cm]
{comastri-andrea_fig_2.ps}
\caption{Scattering fraction vs reflection component intensity. Red points are
from Eguchi et al. (2009), blue points refer to the 3 objects in the present sample
for which the scattering fraction could be estimated.}
\end{figure}

\begin{table}[h]
\caption{Suzaku sample }
\begin{center}
\begin{tabular}{lcccc} \hline\hline\\[-6pt]
Source Name & $N_H$ (cm$^{-2})$ & $f$(\%) & $R$   & K$\alpha$ EW (eV)\\   \hline
ESO137-G34  & 10$^{25}$ &       ...     &  ...    & 1350      \\
ESO323-G32  & 10$^{25}$ &       ...     &  ...    & 2200     \\
NGC 5728    & 1.5 10$^{24}$ &   1.5     &  0.35   & 1040     \\
ESO263-G13  & 2.6 10$^{23}$ &   0.8     &  $<$0.6 &   80     \\
NGC 4992    & 5.5 10$^{23}$ &   0.2     &  2.2    & 450      \\   \hline
\end{tabular}
\end{center}
\end{table}

\subsection{Soft X--ray emission lines}

An equally good description in terms of spectral fits quality is obtained
assuming that the soft X--ray emission is due to a blend of emission lines
which are tipically resolved in good signal to noise
XMM--{\it Newton} reflection grating spectra (Guainazzi \& Bianchi 2007).
While photoionization codes (e.g. {\sc xstar}) are usually adopted to model high resolution
soft X--ray spectra, the statistical quality  of the available  {\sc ccd}
data is such that a simple parameterization, in terms of a power law plus narrow Gaussian
lines, is well suited for the present purposes.
The power law slope is free to vary and is independent from that of the obscured
high energy continuum to possibly account for ``residual" emission
from thermal gas. Individual, narrow Gaussian
lines are then added at the best fit energy
of the most common transitions observed in other Seyfert galaxies.
The line energy and intensity are free parameters of the fit.
The number of lines considered for each source depends
from the actual counting statistics. The ionized Oxygen
{\sc oviii} line at 0.65 keV and {\sc neix} at 0.92 keV are present in
all the objects. A large fraction of the soft X--ray flux in NGC 4992,
the source with the lowest scattering fraction in our sample (Fig.~3), can be accounted for
in terms of line emission. The richest line spectrum is that of NGC 5728
where nine indipendent lines (from Oxygen to Calcium) can be fitted (Fig.~4).

\begin{figure}[t]
\centering
\psbox[xsize=8cm,rotate=r]
{comastri-andrea_fig_3.ps}
\caption{The unfolded spectrum of NGC 4992, fitted with an absorbed power
law plus a Iron line at high ($>$ 3--4 keV) energies. The weak soft X--ray spectrum
can be fitted with a power law and a sum of Gaussian lines.}
\end{figure}

The best fit parameters of the hard X--ray continuum are fully consistent
with those obtained with a partial covering fit.

\begin{figure}[t]
\centering
\psbox[xsize=8cm,rotate=r]
{comastri-andrea_fig_4.ps}
\caption{The unfolded spectrum of NGC 5728, fitted with the same model adopted
for NGC 4992. The nuclear source is obscured by  Compton Thick  gas and
strong K$\alpha$ and K$\beta$ iron lines are detected}
\end{figure}

Both models provide a good description of the observed spectra and
it is not possible, with the available counting statistics, to choose a
``best fit" model. The explanation of the soft X--ray excess as
a blend of unresolved (at the {\sc ccd} resolution) lines from a photoionized
plasma is consistent with previous X--ray observations
of nearby, obscured AGN (Guainazzi \& Bianchi 2007).

\section{The space density of obscured AGN}

Hard X--ray selection is, in principle, almost unbiased against heavy obscuration
and thus considered to be well suited to estimate the intrinsic
absorption distribution of AGN and, in particular, the relative
fraction of Compton Thick  AGN in the local Universe.
The large majority of {\sc integral/ibis} and {\sc swift/bat}
flux limited samples of bright AGN were observed by XMM and {\it Chandra}
and were the subject of follow--up dedicated programs with {\it Suzaku}.
Surprisingly, only the already known Compton Thick AGN were recovered
by the above mentioned hard X--ray surveys and no examples
of newly discovered Compton Thick sources are reported in the literature.
As a consequence, the relative fraction of Compton Thick AGN falls
short by a factor of about 2 than that predicted by the Gilli et al. (2007)
XRB synthesis model at the {\sc integral} and {\sc swift} limiting
fluxes.
A relatively low contribution of CT AGN is predicted by the XRB
synthesis model of Treister et al. (2009). By fixing the low redshift
CT AGN fraction to that observed by hard X--ray surveys, they conclude that
the total contribution of Compton Thick obscured accretion to the XRB is
of the order of 10\%, to be compared with about 25\% of Gilli et al. (2007).
\par\noindent
It is important to stress that the observed fraction in the local Universe
is estimated using a relatively low number of sources and indeed the associated errors
are large. Nevertheless, it seems that present observations are favouring
a relatively low space density of heavily obscured sources.
It is possible to reduce the discrepancy between the
the original Gilli et al. (2007) model predictions and the observations
assuming a lower ratio (0.3 instead of 1) between obscured and unobscured, luminous
($L_X > 10^{44}$ erg s$^{-1}$) AGN (upper envelope
of the shaded region in figure 5). A better agreement (lower envelope of the shaded region in
figure 5) is obtained assuming a slightly
different $N_H$ distribution. More specifically, the space density of transmission
dominated Compton thick AGN, detectable by surveys above 10 keV, is reduced and, at the
same time, that of Compton thin and reflection dominated sources is increased
in such a way to keep approximately constant the total number of obscured AGN
in the model and to fit the XRB spectrum.

\begin{figure}[t]
\centering
\psbox[xsize=7cm]
{comastri-andrea_fig_5.ps}
\caption{ The fraction of Compton thick AGN as a function of the X--ray flux. The dashed line and the
yellow shaded region correspond to the predictions of Treister et al. (2009) and Gilli et al. (2007)
respectively for the 20--40 keV band, while the points with error bars are the observational
estimates from
BAT and IBIS. The blue shaded region and the points at low X--ray fluxes correspond to the Gilli et al
predictions in the 2--10 keV band and the ``observed" fraction in various surveys as labeled,
respectively.}
\end{figure}

\par\noindent
The discrepancy between the two models is now negligible at bright
X--ray fluxes, while it remains significant at lower fluxes and higher
redshifts.

Deep {\it Chandra} surveys have unveiled several examples of heavily obscured
candidate Compton Thick AGN (Tozzi et al. 2006; Georgantopoulos et al. 2009).
At the face value the observed fraction is consistent with the Gilli et al. (2007)
predictions in the 2--10 keV band, though the statistical fluctuations, due to the
low number of objects and systematics errors associated to the column density determination,
may well be larger than those plotted in figure 5.

A significant improvement in the study of Compton thick AGN at $z\sim$ 1, and
possibly beyond is expected by the ongoing ultra--deep (2.5 Msec) XMM--{\it Newton}
survey in the CDFS.
Thanks to the large throughput of XMM it will be possible to obtain good quality
X--ray spectra for a sizable number of sources.
The {\it pn} spectrum of a candidate CT AGN at $z=1.53$ in the
Tozzi et al. (2006) sample,
obtained with about half of the total final exposure, is reported in figure 6.
A strong (EW of about 1 keV) K$\alpha$ line is detected on top of a very
hard continuum and is interpreted as the signature of Compton Thick obscuration.

Indirect searches for Compton Thick AGN at $z \simeq$ 2, using a mid--IR optical
color selection (Daddi et al. 2007; Fiore
et al. 2008; 2009), seem to suggest that heavily obscured accretion
at high redshift may be common, in line with the theoretical expectations.
However the possible contamination from starbust galaxies is a serious issue
and the determination of the space density of Compton thick AGN at high--$z$
is still affected by large uncertainities. Forthcoming {\sc herschel} observations
will allow to disentangle starburst from nuclear emission, extending
previous studies with {\sc spitzer} to higher redshifts.

The direct detection of high redshift, heavily obscured Compton Thick AGN
and the statistical study of their properties (luminosity function and evolution)
cannot be achieved without imaging observations in the hard X--rays.
In this respect, a major step forward is expected by future planned missions
such as {\sc astro-h} and {\sc nustar}.

\begin{figure}[t]
\centering
\psbox[xsize=8cm]
{comastri-andrea_fig_6.ps}
\caption{The XMM-Newton spectrum of a candidate
Compton Thick AGN in the CDFS (for a total exposure time of $\sim$ 1.5 Ms).}
\end{figure}

\vspace{1pc}

\section*{References}

\re
Antonucci R.R.J. \& Miller J.S. 1985 ApJ. 297, 621

\re
Beckmann V. et al. 2006 ApJ. 638, 642

\re
Comastri A. et al. 2009 ApJ. submitted

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Daddi E. et al. 2007 ApJ. 670, 173

\re
Eguchi S. et al. 2009 ApJ. 696, 1657

\re
Fiore F. et al. 2008  ApJ. 672,, 94

\re
Fiore F. et al. 2009  ApJ. 693, 447

\re
Georgantopoulos I., et al. 2009 arXiv0909.0224

\re
Gilli R. et al. 2007 AA 463, 79

\re
Guainazzi M. \& Bianchi S. 2007 MNRAS 374, 1290

\re
Hopkins P.F. et al. 2008 ApJS. 175, 356

\re
La Franca F. et al. 2005 ApJ. 635, 864

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Maiolino R. et al. 2007 AA 468, 979

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Markwardt et al. 2005 ApJ. 633, L77

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Simpson C. 2005 MNRAS 360, 565

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Tozzi P. et al. 2006 AA 451, 457

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Treister E. \& Urry C.M. 2006 ApJ. 652, 79

\re
Treister E. et al. 2009 ApJ. 696, 110

\label{last}

\end{document}


text:
\section{Introduction}

Ultracold atoms in optical lattices \cite{Lewenstein2007,Jaksch2005,Bloch2012}
are versatile systems for studying nonequilibrium physics
\cite{Greiner2002,Schutzhold2006,Fischer2008,Pollack2010,Simon2011,Lin2011,Daley2012,Hung2012
, particularly for making time-resolved measurements of system observables.
For example, lattice collapse-and-revival experiments
\cite{Greiner2002,Will2010,Anderlini2006,Sebby2007} follow atom populations in
the $k=0$ quasimomentum of nonequilibrium Bose gases versus system evolution
time. Our goal is to extend
the power of these experiments to enable time-resolved measurements of a
greater variety of system observables and Hamiltonians. A second goal is to
use nonequilibrium collapse-and-revival dynamics to investigate the physics of
interaction-based (or nonlinear) quantum metrology, which exploits multibody
dynamics to characterize a system and its parameters
\cite{Boixo2007,Boixo2008,Choi2008,Liu2010,Grond2011,Napolitano2011,Giovannetti2011,Javanainen2012,Benatti2011
.

In this paper, we design an interaction-based interferometer for making
time-resolved measurements of quadrature operators \cite{Walls2008} of
nonequilibrium matter fields in optical lattices. Measurement of quantum-field
quadratures $X_{k}=(e^{-i\zeta_{k}}A_{k}+e^{i\zeta_{k}}A_{k}^{\dagger
)/2,$where $A_{k}$ annihilates atoms with quasimomentum $k$ and spin state
$a$, and $\zeta_{k}$ is a tunable phase, can provide direct determination of
the order parameter for a Bose gas \cite{Kitagawa2011}, as opposed to
inferring it from density measurements \cite{Pethick2008,Pitaevskii2003}. The
technique, analogous to homodyne detection, works as follows. Exploiting the
internal state structure of ultracold atoms, we split a superfluid into two
subsystems---the arms of the interferometer---consisting of atoms in different
spin states and lattice sites. A Feshbach resonance \cite{Chin2010} turns off
atom-atom interactions in one spin subsystem, making it a well-characterized
reference \textquotedblleft beam,\textquotedblright\ while atoms in the other
subsystem undergo nonequilibrium dynamics for a variable hold time.
Interfering the subsystems via a second beam-splitting operation, spatial
quadrature observables are obtained by measuring spin populations.

As a second, distinct application of the interferometer, we show that
atom-atom interaction strengths can be determined from the frequencies of
matter-wave collapse-and-revival oscillations with \textquotedblleft
super-Heisenberg\textquotedblrigh
\ \cite{Boixo2007,Boixo2008,Choi2008,Liu2010}\ scaling $\bar{n}^{-3/2}$ in the
mean number of atoms per lattice site $\bar{n},$ and standard quantum limit
(SQL) scaling $M^{1/2}$ in the number of lattice sites $M$. In contrast, we
find that the optimal scaling from conventional collapse-and-revival dynamics,
which do not exploit entanglement of motional and spin degrees of freedom, is
$\bar{n}^{-3/4}.$ Our analysis requires $M\gg1$ and for realistic lattice
systems $\bar{n}\lesssim5;$ consequently, the scaling in total atom number
$N=M\bar{n}$ is below the Heisenberg bound. Nevertheless, we find that
measurements testing the predicted scaling behaviors for interaction-based
metrologies should be possible in a lattice system using quadrature
measurements. We note that measuring the quadratures of nonequilibrium lattice
fields, our first goal, is less experimentally demanding than determining
interaction strengths with optimal precision. Recently, super-Heisenberg
scaling has been realized in another system \cite{Napolitano2011}.

In the following, we first describe the interferometric technique and propose
implementations. We next analyze the special case of atoms held in a deep,
cubic lattice with negligible tunneling, and obtain analytic predictions for
the time evolution of both quadrature observables and their fluctuations. We
emphasize, however, that the technique can provide time-resolved quadrature
measurements for more complicated Hamiltonians and lattice geometries (e.g.,
cubic, honeycomb, superlattices
\cite{Sebby2006,Vaucher2008,Barmettler2008,Assam2010}), including systems with
tunneling, spin-orbit couplings using artificial gauge fields \cite{Lin2011},
and higher-band effects \cite{Muller2007,Bakr2010,Soltan2012}, and a method
for studying quench dynamics \cite{Zurek2005,Cucchietti2007,Chen2011}. Related
theoretical developments on collapse-and-revival physics can be found in
\cite{Buchhold2011,Fischer2012,Gramsch2012,Daley2012}.

\section{Interferometry technique}

Figure~\ref{sketch} shows a schematic of the quadrature interferometer. First,
a superfluid of two-component bosonic atoms, with spin states $\left\vert
a\right\rangle $ and $\left\vert b\right\rangle ,$ are prepared in an optical
lattice with all atoms in state $\left\vert b\right\rangle $. Interactions
between $b$-state atoms ($b$ atoms) are then slowly turned off with a Feshbach
resonance, such that the initial interferometer state is a coherent state with
$k=0$ quasimomentum in the lowest Bloch band of an optical lattice potential.
We therefore have $|\psi_{k=0}\rangle=\exp(\gamma B_{k=0}^{\dagger
-\gamma^{\ast}B_{k=0})|0\rangle$, where $|0\rangle$ is the empty lattice,
$B_{k}$ annihilates $b$ atoms with quasimomentum $k$, and the mean total atom
number is $\bar{N}=|\gamma|^{2}$. We can re-express this as the separable
state $|\psi_{k=0}\rangle=\prod_{i=1}^{M}\exp(\beta b_{i}^{\dagger
-\beta^{\ast}b_{i})|0\rangle$, where $B_{k=0}=\sum_{i}b_{i}/\sqrt{M}$ and
$b_{i}$ annihilate $b$ atoms in the lowest Wannier function of lattice site
$i$. The mean atom number per lattice site is ${\bar{n}}=|\beta|^{2}=\bar
{N}/M$. For notational simplicity, we suppress the band index and vectorial
nature of $i$ and $k$. As an aside, assuming an initial Fock state
$(B_{k=0}^{\dag})^{N}\left\vert 0\right\rangle /\sqrt{N!}$ (e.g., see
\cite{Schachenmayer2011}) leads to the same interferometer results found below
with finite-size corrections of order $M^{-3/2}$.

\begin{figure}[ptb]
\begin{center}
\includegraphics[scale=1.0,trim=0 0 0 0,angle=0,clip]{NLI7} \vspace*{-28pt}
\end{center}
\caption{(Color online) Schematic of quadrature operator interferometer for
two-state atoms held in $M$ optical lattice sites. Time evolution is left to
right. The left-most box represents initial state preparation. Solid and
dashed lines correspond to atoms in lattice sites $i$ and Bloch states with
quasimomenta $k$, respectively. Bold-red and thin-blue lines are atoms in spin
states $|a\rangle$ and $|b\rangle$, respectively. Green lines indicate
superposition states. Arrows specify spin states. Gray and red boxes
correspond to unitary operations described in the text. Detectors (brown) at
right measure population differences between spin states at quasimomenta $k$.

\label{sketch
\end{figure}

The lattice depth is next increased to turn off tunneling without exciting
atoms to higher bands, and the Wannier function can be approximated by the
energetically lowest local orbital in a site. At this point atoms neither
tunnel nor interact. A $\pi$ pulse between internal states $a$ and $b$ is then
applied to half of the lattice sites, transforming the system state to
$\prod_{i=1}^{M/2}\exp(\beta a_{i}^{\dagger}-\beta^{\ast}a_{i})\otimes
\prod_{j=M/2+1}^{M}\exp(\beta b_{j}^{\dagger}-\beta^{\ast}b_{j})|0\rangle$,
where $a_{i}$ annihilates $a$ state atoms ($a$ atoms) in the lowest orbital of
site $i$. The $\pi$ pulse is the first beam splitter of the interferometer,
creating two arms with $a$ and $b$ atoms, respectively.

After the $\pi$ pulse, atoms are held for a variable evolution time $t$. The
$a$ atoms interact and undergo nonequilibrium dynamics, while use of the
Feshbach resonance ensures that $b$ atoms remain noninteracting and provide a
well-characterized reference \textquotedblleft beam\textquotedblright\ state.
Spatial separation between $a$ and $b$ atoms is required because it
effectively turns off their mutual interactions. After this point, we can
change the system dynamics to a \textquotedblleft nontrivial\textquotedblrigh
\ Hamiltonian, as long spatial separation of $a$ and $b$ atoms is maintained.
For example, tunneling can be turned back on, the lattice geometry dynamically
transformed, or external fields applied.

The second beam splitter combines two operations. First, a Fourier transform
on atom spatial modes is applied, transforming site indices $i$ into
quasimomentum indices $k$. This is achieved by releasing atoms from the
lattice followed by (time-of-flight) expansion until all atoms have spatially
overlapping wave functions. Second, a $\pi/2$ pulse with tunable phase $\chi$
is applied to the internal states of all atoms, driving $|a\rangle
\rightarrow(e^{i\chi/2}|a\rangle+e^{-i\chi/2}|b\rangle)/\sqrt{2}$ and
$|b\rangle\rightarrow(-e^{i\chi/2}|a\rangle+e^{-i\chi/2}|b\rangle)/\sqrt{2}$.

Finally, we measure the observable ${S}_{k}=A_{k}^{\dagger}A_{k
-B_{k}^{\dagger}B_{k}$ by detecting the difference between the number of $a$
and $b$ atoms in quasimomentum $k$ $\emph{after}$ the second beam splitter.
Switching to the Heisenberg picture and using $A_{k}\rightarrow(e^{i\chi
/2}A{_{k}+e^{-i\chi/2}}B{_{k}})${$/\sqrt{2}$} and $B${$_{k}\rightarrow
(-e^{i\chi/2}A_{k}+e^{-i\chi/2}B_{k})/\sqrt{2}$ at the second beam splitter,
we find }that {the observable is equivalent to $S_{k}=$}$A{_{k}^{\dagger
}B{_{k}e^{-i\chi}+}B{_{k}^{\dagger}}A{_{k}e^{i\chi}}$ with expectation values
{taken with }respect to wave functions just \emph{before} the second beam splitter.

Because the many-body wave function is separable in the internal atomic states
before the $\pi/2$ pulse, it follows that $\langle S_{k}\rangle=\langle
A_{k}\rangle\langle B_{k}^{\dagger}\rangle{e^{-i\chi}}+\langle A_{k}^{\dagger
}\rangle\langle B_{k}\rangle{e^{i\chi}}$. The noninteracting reference atoms
give $\langle B_{k}\rangle=\beta g_{k}$ with $\beta=\sqrt{\bar{n}}e^{i\theta}$
and the complex-valued $g_{k}$ depends only on lattice geometry and
distribution of $b$ atoms. In particular, $g_{k=0}=\sqrt{M}/2$ for any lattice
geometry and atom distribution. We therefore find
\begin{equation}
\langle S_{k}\rangle=2\sqrt{\bar{n}}|g_{k}|\langle X_{k}\rangle, \label{Eq1
\end{equation}
where $X_{k}$ is the quadrature observable with tunable phase $\zeta_{k
=\arg(e^{i(\theta+\chi)}g_{k}).$ In contrast, regular collapse-and-revival
experiments \cite{Greiner2002,Will2010,Sebby2007,Anderlini2007}, using atoms
in only one spin state, measure the quasimomentum distribution $R_{k
=A_{k}^{\dagger}A_{k}$.

\section{Implementations}

Realization of the interferometer requires two internal atomic states that
have little or no two-body collisional loss, while allowing for tuning of
$b+b$ collisions to turn off interactions. We suggest the two energetically
lowest hyperfine states of bosonic alkali-metal atoms as only weak magnetic
dipole-dipole and second-order spin-orbit interactions can cause inelastic
loss. Candidates are $^{87}$Rb near the narrow resonance at a magnetic field
$B$ of 100.7 mT \cite{Marte2002} and $^{39}$K near a much broader resonance at
40.2 mT \cite{DErrico2007}, the latter requiring less stringent field control.
Other examples can be found in \cite{Chin2010}. Figure~\ref{preparation}(a)
shows the scattering length $a_{s}$ of the $^{39}$K resonance as a function of
$B$ and indicates two field values where one of the two hyperfine states has
zero scattering length and, hence, does not interact.

The initial state of the interferometer is prepared by starting with a
(superfluid) Bose condensate of \emph{interacting} $b$-state atoms in a weak
trap with level spacing $\hbar\omega_{\mathrm{w}}$, typically harmonic with
$\omega_{\mathrm{w}}/2\pi=1-100$ Hz; $\hbar$ is Planck's reduced constant.
After adiabatically turning on the optical lattice, the system can be
described by the single-band Bose-Hubbard Hamiltonian
\cite{Fisher1989,Freericks1994,Jaksch1998,vanOosten2001} with tunneling energy
$J$ and atom-atom interaction strength $U_{bb}$ proportional to $a_{s}$. In a
simple cubic lattice, a superfluid ground state requires $J\gg U_{bb
{/(5.83\times6)}$ at the end of the adiabatic ramp (the numerical factor
follows from a three-dimensional mean-field calculation). Typically,
$U_{bb}/h\sim1$ kHz away from Feshbach resonances. Due to atom-atom
interactions, at this stage the superfluid is only an approximate coherent state.

Figure~\ref{preparation}(b) shows the next step in initial state preparation.
Slowly turning off $U_{bb}$ at fixed $J$ using a Feshbach resonance over
timescale $\tau_{\mathrm{int}},$ the ground state approaches the desired
coherent state of noninteracting $b$ atoms. Choosing $\tau_{\mathrm{int}
\gg2\pi/\omega_{\mathrm{w}}$ minimizes excitation from zero quasimomentum.
After interactions are ramped-off, tunneling is suppressed by increasing
lattice depth with time scale $\tau_{\mathrm{tun}}$. Choosing $\tau
_{\mathrm{tun}}\ll2\pi/\omega_{\mathrm{bg}}$, where $\hbar\omega_{\mathrm{bg
}$ is the lattice band gap (typically $\omega_{\mathrm{bg}}/2\pi=10-100$ kHz),
suppresses excitation to higher bands.

\begin{figure}[ptb]
\begin{center}
\includegraphics[scale=1,trim=0 0 0 0,angle=0,clip]{ISP4} \vspace*{-17pt}
\end{center}
\caption{(Color online) (a) Scattering lengths $a_{s}$ of $^{39}$K collisions
versus magnetic field $B$ near a Feshbach resonance located at $B\approx40$
mT. Curves are labeled by $|fm\rangle=|11\rangle$ and $|10\rangle$, the
energetically lowest hyperfine channels of $^{39}$K. Two operating points for
the interferometer are shown by black-dotted lines (here $1a_{0}=0.0529$ nm).
(b) The interferometer initial state, a coherent state of noninteracting
atoms, is created starting from a superfluid of interacting atoms by using a
Feshbach resonance to slowly turn off the interaction strength $U_{bb}$,
followed by faster turn off of the tunneling rate $J$. See text for time and
energy scales. Panels (c) and (d) show two possible atom distributions after
the site-specific $\pi$ pulse. In panel (c) atoms in state $a$ (red spheres)
and $b$ (blue spheres) are located in the top and bottom half, respectively.
In panel (d) planes of state $a$ and $b$ atoms alternate.
\label{preparation
\end{figure}

The first beam splitter, a spatially selective $\pi$ pulse, can be
implemented, for example, by illuminating only the top half of the lattice
sites [see Fig.~\ref{preparation}(c)] with lasers driving optical Raman
transitions via off-resonant transitions to electronically excited states. The
interaction strength for $a$ atoms is $U_{aa}.$ This approach is limited by
the boundary over which the Raman laser intensities go to zero, which is no
sharper than the laser wavelength. Uncertainty in the fraction of $a$ versus
$b$ atoms leads to only small corrections of order $M^{-3/2}.$ An alternative
approach uses dynamically transformable lattices while simultaneously flipping
selected spin states. For example, Refs.~\cite{Anderlini2006,Anderlini2007}
used spin-dependent lattices to transform from a single- to double-well
lattice where the spin state of atoms alternates with lattice site. In
principle, alternating planes of $a$ and $b$ atoms can be created [see
Fig.~\ref{preparation}(d)] and our interferometric techniques used to study
correlations in two-dimensional many-body physics.

The second beam splitter involves release and expansion from the lattice
followed by a global $\pi/2$ pulse on the internal state of all atoms. The
latter operation requires no spatial selectivity and can be implemented with
Raman transitions, or microwave and radio-frequency radiation. The phases
$\pi/2$ and $\chi$ are controllable by choosing the polarization of the
radiation relative to the applied magnetic field; the required techniques have
been demonstrated in atomic clock and ion trap quantum computing
implementations \cite{Haffner2008}. Requirements on expansion time are
analyzed in \cite{Toth2008}, and we estimate that $\sim20$ ms is sufficient
for operation of the interferometer.

Finally, there are corrections due to interactions after atoms are released
from the lattice. Extending \cite{Toth2008}, we model the dominant effect of
\emph{coherent }collisions by a time-varying $U_{aa}$ proportional to the
inverse volume of the expanding local orbital. This leads to a small
additional evolution time of order $1/\omega_{\mathrm{bg}},$ typically
$\sim10$ $\mu$s, and independent of hold time $t$, and does not degrade the
performance of the interferometer.

Momentum changing, \emph{incoherent }collisions can also occur between atoms
originating from either the same or different lattice sites. We expect the
effect of these thermalizing collisions to be much smaller than those of
coherent collisions not only because of the rapidly decreasing density, but
also due to the rapidly decreasing relative collisional energies
($\approx\hbar\omega_{\mathrm{bg}}$ at the moment of release) during
expansion. The resulting lack of significant thermalization is supported by
existing collapse-and-revival experiments
\cite{Greiner2002,Will2010,Anderlini2006,Sebby2007}, which have demonstrated
the momentum distribution measurements needed for our proposed technique.

We also need to minimize effects from interactions between $a$ and $b$ atoms.
This can be achieved by delaying the $\pi/2$ pulse by $\gtrsim1/\omega
_{\mathrm{bg}}$ after release, but before significant phase shifts arise from
a combination of any magnetic field inhomogeneities and the differential
magnetic moment of the two spin states. Effects of field gradients after the
$\pi/2$ pulse should then only shift the relative location of the $k=0$ modes
for $a$- and $b$-atom clouds.

\section{Nonequilibrium dynamics}

The factorization property leading to Eq.~\ref{Eq1} holds even if tunneling
and other modified dynamics are induced for the interacting $a$ atoms after
the first beam splitter, making it possible to use the technique to measure
the quadrature of nonequilibrium fields for many types of Hamiltonians (e.g.
systems with spin-orbit couplings, different lattice geometries, or applied
fields). Below, we analyze the interferometer and predict the quadrature
evolution for the relatively simple evolution of the Bose-Hubbard model
without tunneling, i.e., $H_{\mathrm{MB}}=(U_{aa}/2)\sum_{i=1}^{M/2
a_{i}^{\dagger}a_{i}^{\dagger}a_{i}a_{i}$. Effective multibody interactions
\cite{Johnson2009,Johnson2012,Bissbort2012} are omitted for simplicity.

We focus on the normalized $k=0$ quasimomentum observable $s_{0}(t)=\langle
S_{k=0}(t)\rangle/\bar{N}$, which is proportional to the order parameter.
Defining single-site quadrature operators $x_{i}=(e^{-i\zeta_{0}
a_{i}+e^{i\zeta_{0}}a_{i}^{\dagger})/2$, we find $s_{0}(t)=\langle
x_{i}(t)\rangle/(2\sqrt{\bar{n}}),$ after expanding the momentum operator in
terms of site operators and realizing that the time evolution is the same for
all sites. For $H_{\mathrm{MB}}$ we derive
\begin{equation}
\langle x_{i}(t)\rangle=\sqrt{\bar{n}}\exp([\cos(\phi)-1]{\bar{n}})\cos
({\bar{n}}\sin(\phi)+\zeta_{0}),
\end{equation}
with phase $\phi=U_{aa}t/\hbar$. Figures~\ref{time}(a) and \ref{time}(b) show
examples of the periodic evolution of $\langle x_{i}(t)\rangle/\sqrt{\bar{n}}$
for several values of $\zeta_{0}$ and $\bar{n}.$ The interferometer makes
possible direct experimental verification of these quadrature dynamics. The
time evolution of $\mathcal{R}_{k=0}/\bar{N}=|\langle a_{i}\rangle|^{2
/{\bar{n}}$ {(assuming large }${M}${),} determined by regular
collapse-and-revival measurements
\cite{Greiner2002,Anderlini2006,Sebby2007,Will2010}, is also shown.
\begin{figure}[ptb]
\begin{center}
\includegraphics[scale=0.34,trim=0 0 0 0,angle=0,clip]{TimeTraceJoined}
\vspace*{-30pt}
\end{center}
\caption{(Color online) Panels (a) and (b) show the normalized single-site
quadrature $\langle x_{i}\rangle/\sqrt{\bar{n}}$ (solid lines) as a function
of evolution time in units of $h/U_{aa}$ for various quadrature phases and
mean atom number $\bar{n}$. The double-sided arrow indicates a possible time
interval for measuring $U_{aa}$. The density operator $|\langle a_{i
\rangle|^{2}/{\bar{n}}$ (dashed line) is only shown for $\bar{n}=4$. Panels
(c) and (d) show the quadrature uncertainties $\sigma_{xi}$ as a function of
evolution time and $\bar{n}$, for $\zeta_{0}=0$ and $\zeta_{0}=\pi/2,$
respectively. The dash-dotted line, labeled SQL, indicates $\sigma_{xi}=1/2$
for a coherent state.
\label{time
\end{figure}

The fluctuations in quadrature observables for nonequilibrium Bose gases in
deep lattices are also interesting. Figures~\ref{time}(c) and \ref{time}(d)
show the analytic time-dependent (single-site) uncertainty $\sigma
_{xi}=\langle(x_{i}-\langle x_{i}\rangle)^{2}\rangle^{1/2}$ as a function of
time. For a coherent state for any $\bar{n}$ and $\zeta_{0},$ $\sigma
_{xi}=1/2$, corresponding to the SQL and shown by horizontal dashed lines;
$\sigma_{xi}<1/2$ indicates quadrature squeezing. For most times and phases
$\zeta_{0},$ the uncertainty is $\sigma_{xi}>1/2$; in fact, a
\textquotedblleft plateau\textquotedblright\ develops for large $\bar{n}$
where $\sigma_{xi}$ approaches $\sqrt{2{\bar{n}}+1}/2$. For times near integer
multiples of $U_{aa}t/h$ and values of $\zeta_{0}$ near zero, however,
$\sigma_{xi}<1/2$, and the fluctuations in $\langle x_{i}\rangle$ are reduced
below the SQL. This is seen in Fig.~\ref{time}(c) for $\zeta_{0}=0,$ where we
find that the minimum value of $\sigma_{xi}$ approaches a constant
$\approx0.29$ for large $\bar{n}.$ In contrast, Fig.~\ref{time}(d) shows that
$\sigma_{xi}\ $for $\zeta_{0}=\pi/2$ is always antisqueezed.

\section{Interaction-based metrology}

In addition to directly measuring field quadratures, the interferometry can be
used to measure interaction strengths, although achieving optimal performance
is experimentally more demanding. The period $h/U_{aa},$ and hence $U_{aa},$
can be obtained by determining times $t_{1}$ and $t_{2}>t_{1}$ at which
$s_{0}(t)$ and its time derivative are the same. Figure~\ref{time}(a) shows an
example where $U_{aa}=h/(t_{2}-t_{1})$. Error propagation gives the fractional
uncertainty
\begin{equation}
\frac{\delta U_{aa}}{U_{aa}}=\frac{\sqrt{2}}{\pi\sqrt{M/2}}\left\vert \frac
{1}{d\langle x_{i}\rangle/d\phi}\right\vert \sqrt{\sigma_{xi}^{2
+\frac{\langle x_{i}\rangle^{2}}{2{\bar{n}}}}\,, \label{eq2
\end{equation}
with expectation values evaluated at phase $\phi_{1}=U_{aa}t_{1}/\hbar$,
assuming equal contributions at $t_{1}$ and $t_{2}$ added in quadrature. The
fractional uncertainty gives SQL scaling $\propto1/\sqrt{M/2}$ in number of
$a$-atom lattice sites, as expected for a site-separable wave function, and we
can view the interferometer as $M/2$ independent probes of $\bar{n}$
interacting atoms. The second term under the square root is due to the
uncertainty in total atom number $\bar{N}$. The uncertainty also scales as
$1/m$ in the number of oscillation periods $m$; for brevity we set $m=1.$

Minimizing $\delta U_{aa}/U_{aa}$ based on measurements of $x_{i}(t)$ involves
a trade-off between identifying optimal measurement times $t_{1}$, $t_{2}$ (or
phases $\phi_{1},\phi_{2}$) that maximize the slope $d\langle x_{i
\rangle/d\phi$ while minimizing $\sigma_{xi}.$ We find that the smallest
(optimal) fractional uncertainty
\begin{equation}
(\delta U_{aa}/U_{aa})_{\text{quad}}=(1/2\pi)M^{-1/2}\bar{n}^{-3/2},
\label{eq3
\end{equation}
is obtained from the quadrature observable with $\zeta_{0}=\pi/2$, and
measurements made when $\langle x_{i}\rangle=0$ and the slope $d\langle
x_{i}\rangle/d\phi$ is maximum. This occurs at phases $\phi_{1}=0$ and
$\phi_{2}=2\pi$, corresponding to times $t_{1}=0$ and $t_{2}=2\pi\hbar/U_{aa
$. Because $\langle x_{i}\rangle=0$ and $\sigma_{xi}=1/2$ at these times, the
result is independent of uncertainty in $\bar{N}$. For other $\zeta_{0},$ the
numerical prefactor in Eq.~(\ref{eq3}) is larger, but the scaling in $\bar{n}$
is the same. As noted in \emph{Implementations}, the small correction from
interactions during expansion can be included in the determination of
$t_{1,2}$.

At optimal measurement times the state is unsqueezed, and the super-Heisenberg
scaling $\bar{n}^{-3/2}$ derives from the nonlinear dependence of $d\langle
x_{i}\rangle/d\phi$ on $\bar{n}.$ Our quadrature interferometer yields the
best possible scaling behavior predicted for interaction-based metrology
schemes based on two-body interactions and nonentangled states
\cite{Boixo2008}. We have also analyzed the minimal uncertainty possible from
conventional collapse and revival measurements of $\langle\mathcal{R
_{k=0}(t)\rangle/\bar{N}$. We obtain the analytic estimate$\ (\delta
U_{aa}/U_{aa})_{\text{conv}}=(2/3)^{3/4}\pi^{-1}M^{-1/2}\bar{n}^{-3/4},$ and
have confirmed it is in good agreement with numerical evaluation. The scaling
is significantly less advantageous than what is possible using the quadrature
interferometry.

We can compare our predictions with conventional collapse-and-revival
experiments. Will \textit{et al.} \cite{Will2010} obtained $\delta
U_{aa}/U_{aa}\approx2\times10^{-2}$ with $\bar{N}=2\times10^{5}$ atoms and
${\bar{n}}=2.5$. For these parameters $(\delta U_{aa}/U_{aa})_{\text{conv
}=3\times10^{-4}.$ The larger experimental uncertainty is likely due to
fluctuations in the total atom number and lattice-laser intensities. For the
quadrature interferometer $(\delta U_{aa}/U_{aa})_{\text{quad}}=1.2\times
10^{-4}$ is, in principle, possible, giving nearly a factor of three
improvement even with ${\bar{n}}=2.5.$ As noted above, we also predict less
sensitivity to fluctuations in $\bar{N}$ using the quadrature technique.

\section{Conclusions}

Our analysis suggests that even for small (realistic) $\bar{n}$ it should be
possible to test predicted scaling behaviors for interaction-based metrologies
in an optical lattice system. There are many interesting directions for future
research. For example, are there interferometer states that give $\bar{n
^{-2}$ scaling per lattice site, as suggested by the analysis in
\cite{Boixo2008}? Can we design implementations where the important
information lies in $k\neq0$ quasi-momenta states? If so, this would be a
major application of our technique toward the study of nonequilibrium
many-body systems. Can we modify the interferometer to measure observables
such as $\langle a_{i}a_{j}\rangle,$ and use entanglement between atoms in
different sites to measure nonlocal field correlations? There are also
practical challenges facing implementations, including the effects of
inhomogeneities, lattice (laser) fluctuations, and limits on both atom number
per site and total atom number. Even with non-optimal measurements, however,
the interferometer technique developed here should expand the toolkit for
studying nonequilibrium dynamics in optical lattices.

\begin{acknowledgments}
P.R.J. and E.T. acknowledge support from the U.S. Army Research Office under
Contract No. 60661PH. P.R.J. also acknowledges computing resources provided by
the American University High Performance Computing System. E.T. acknowledges
support from a National Science Foundation Physics Frontier Center. Finally,
we thank Khan Mahmud, Lei Jiang, and Nathan Harshman for valuable discussions.
\end{acknowledgments}