text:
\section{Introduction}
Among the capabilities enabled by electromagnetically induced transparency (EIT)  \cite{Boller,Fleischhauer2005} or related
phenomena are high precision magnetometery,
lasing without inversion, and slowing
\cite{Hau1999} and stopping of light pulses
\cite{Liu2001,Phillips2001}. These possibilities opened new
avenues for optical information storage and quantum information
processing. Recent experiments based on dynamic EIT have
demonstrated the reversible mapping of single-photons or qubits
\cite{Matsukevich2005,Eisaman2005,Choi2008} and of quantum
continuous variables \cite{Honda2008,Appel2008,Cviklinski2008}.
These seminal demonstrations spurred intense experimental and
theoretical efforts to improve the efficiency of such processes
and extend them to new enabling photonic technologies.

An important effort concerns indeed the modeling of the EIT
process in the non-ideal case. The EIT configuration is usually
modeled by a generic $\Lambda$-type three-level system,
 most relevant for quantum memory schemes \cite{Fleischhauer2005}: two atomic
ground states are connected to an excited state via two optical
fields, a probe and a control field. However, in many optically
dense media, the relevant energy structure is more complex and can
strongly modify the EIT features. A typical case is the use of
ensembles of alkali-metal atoms in which experiments have been
most performed \cite{Lvovsky2009,Hammerer2010}. The hyperfine
interaction in the excited state introduces several levels, which
can simultaneously participate in the coherent interaction. The
deviation from the $\Lambda$-type approximation can be very
significant when the inhomogeneous broadening is comparable with
the separation between these excited levels, such as for example
in the D$_2$-line of atomic cesium at room temperature. The
observed transparency is generally lower than predicted
\cite{Cviklinski2008,Akulshin1998} and can eventually disappear
for large broadening \cite{Li2009}. The broadening also leads to a narrowing of the transparency window \cite{Ye2002,Hossain2009,Figueroa2006}. Various
numerical analysis have investigated particular regimes, such as
double-$\Lambda$ system \cite{Cerboneschi1996} or off-resonant
Raman transition in a broadened medium \cite{Sheremet2010}.
However, to date no full study of EIT in inhomogeneously broadened
medium with multiple excited levels has been performed.

In the present paper, we report measurements that provide a detailed
picture of EIT in a Doppler broadened medium. In agreement with
the general model recently developed in Ref. \cite{Oxy2011}, we
evidence the process leading to a reduced transparency and we
experimentally demonstrate how to mitigate this effect. Our
observations are made possible by identifying atoms from specific
velocity classes that absorb the light in the process and by then
reshaping accordingly the atomic velocity distribution. This
procedure enables to recover a significant transparency.

\section{Effect of inhomogeneous broadening on
the EIT features}
 In a $\Lambda$-type system, the
susceptibility of the medium is strongly modified when a driving
field is applied to one of the transition. In the absence of
broadening, the transmission of a probe field exhibits two symmetrical absorption peaks as a function of
its detuning from the resonance, defining a
transparency window at resonance. In an inhomogeneously broadened
medium, the absorption spectrum of the various atoms differs from
this description as they are involved in off-resonant
processes, drastically modifying the susceptibility and the EIT spectrum.

\begin{figure*}[htpb!]
\includegraphics[width=1.5\columnwidth]{figure1}
\caption{EIT features in a $\Lambda$ and in a multi-level system. The calculated probe absorptions are displayed for atoms
with different velocities  given by the Doppler detunings
$\Delta_{Doppler}$, as a function of the detuning $\Delta_{2ph}$. Panel (a)
corresponds to a $\Lambda$-scheme and shows a preserved transparency.
Panel (b) takes into account an additional excited state, leading
to a vanishing transparency. The dotted line indicates the
center of the EIT window for atoms with $\Delta_{Doppler}$=0. The
integrated absorption is obtained for a Gaussian velocity
distribution of 160 MHz half-width (thermal distribution for Cesium at 300 K). The control Rabi
frequency is $\Omega=2.3\gamma $, where $\gamma $ is the
natural linewidth.} \label{schema_3D}
\end{figure*}

Specifically, in a Doppler broadened medium, the atoms are distributed over a wide range of velocity classes. For an atom
moving with velocity $\textbf{v}$ and copropagating control and
probe fields, the two laser frequencies are Doppler shifted in the
atom rest frame by approximately the same detuning
$2\pi.\Delta_{Doppler}=-\textbf{k}\cdot\textbf{v}$ where
$\textbf{k}$ is the wave-vector of the fields. The two-photon
detuning, defined as
$2\pi.\Delta_{2ph}=\omega_{probe}-\omega_{control}+\omega_{gs}$
where $\omega_{probe}$ and $\omega_{control}$ are the field
frequencies and $\omega_{gs}$ the splitting between the two ground states, does not depend on the velocity either.
Figure \ref{schema_3D}(a) gives the absorption as a function of
the two-photon detuning for different Doppler shifts when the
control field is resonant for atoms at rest. We
have considered the case of velocities opposite to the
laser propagation direction ($\Delta_{Doppler}>0$). While for
atoms at rest one can observe the usual Autler-Townes doublet, the
spectrum is modified when the Doppler shift increases, i.e. the
two absorption peaks are not symmetrical anymore. The peak with
$\Delta_{2ph}<0$ corresponds to the one-photon absorption
resonance and the second peak ($\Delta_{2ph}>0$) corresponds to
the Raman absorption process, and it is Stark shifted from the
zero two-photon detuning. When the Doppler detuning increases,
this Raman peak gets closer to $\Delta_{2ph}=0$ without reaching
this point. As a result, transparency is preserved at the
zero two-photon detuning for all the velocity classes. The
integration over the whole distribution thus preserves the transparency at resonance but
leads to a reduction of the transparency window width.

When one takes into account the presence of other levels,
 additional Stark shifts appear that move the Raman resonance from the
position described in the $\Lambda$ configuration.  Based on the
model developed in \cite{Oxy2011}, Fig. \ref{schema_3D}(b) shows
the case of two excited levels, $|e\rangle$ and $|e'\rangle$. The
strong modification of the susceptibility can be understood as
follows. In this scheme, two velocity classes now see the control
field on resonance but for different levels: $|e\rangle$ for zero
Doppler shift and $|e'\rangle$ for a shift equal to the separation
between the two levels \textbf{$\omega_{e'e}$}. These two classes
exhibit a quasi-symmetrical Autler-Townes doublet centered
close to their respective atomic transition, i.e. on the
zero two-photon detuning shown on Fig. 1(b). For atoms with
intermediate velocities, the Raman absorption undergoes Stark
shifts due to the two excited states and eventually crosses the
transparency window of atoms with zero Doppler shift, as
illustrated in Fig. \ref{schema_3D}(b). While transparency was
always preserved at resonance in the $\Lambda$ model, here there
is no longer any value of the detuning for which atoms are
transparent independently of their velocity. Consequently, if the
broadening is comparable with the hyperfine splitting, the
integration over all the velocity classes can result in a total
disappearance of the EIT, as shown here. We find that in our case the atomic velocity
classes to be removed for optimal EIT recovery correspond to
Doppler detunings 35MHz$\lesssim\Delta_0\lesssim$45MHz (Appendix A). By excluding these specific atoms from the
interaction process, the EIT can be recovered, as we will show.

\begin{figure*}[htpb!]
\includegraphics[width=1.5\columnwidth]{figure2}
\caption{The transmission of a cesium vapor is probed by
scanning the frequency of a weak $\sigma^-$-polarized probe field.
The strong $\sigma^+$-polarized control field is kept on resonance
with the $|s\rangle$ to $|e\rangle$ transition. A repumper beam
enables to efficiently prepare the atoms in the $|g\rangle$ ground
state. Two  $\sigma^+$-polarized depumpers beams can be used to
burn holes in the velocity distribution in order to exclude atoms
with specific Doppler shifts from the interaction process. PBS and NPBS : polarizing and
non-polarizing beam splitter, QWP: $\lambda/4$ plate.}
\label{schema_manip}\vspace{-0.4cm}
\end{figure*}

\section{Experimental setup and results}
The experiment is sketched in Fig.
\ref{schema_manip}. The optically dense medium is obtained from a vapor of $^{133}$Cs heated at 35$^{\circ}$C in a
paraffin-coated cylindrical glass cell (3 cm long and 3 cm in
diameter). At this temperature, the Doppler broadening reaches a
half-width equal to 160 MHz. The cell is placed in a longitudinal
magnetic field produced by sets of coils and the system is
enclosed into a magnetic shield. The scheme of the interaction
is given in the inset of Fig. \ref{schema_manip}. The
two ground states $|s\rangle$ and $|g\rangle$ are the two Zeeman states
$|6S_{1/2},F=3,m=1\rangle$ and $|6S_{1/2},F=3,m=3\rangle$ separated by 1.25 MHz. The excited
state $|e\rangle$ is the Zeeman state $|6P_{3/2},F=2,m=2\rangle$. As explained
previously, this simple $\Lambda$ system is strongly influenced by
the other excited states, i.e. $|6P_{3/2},F=3\rangle$ and $|6P_{3/2},F=4\rangle$ which are
respectively 151 MHz and 352 MHz from the $|e\rangle$ state. The control
field is $\sigma^{+}$-polarized and resonant with the transition
$|s\rangle \rightarrow |e\rangle$ for non-moving atoms, while the probe field is
$\sigma^{-}$-polarized and addresses the transition $|g\rangle \rightarrow
|e\rangle$.

The experiment is performed in the continuous-wave regime. The
control field tends to pump all the atoms into the Zeeman sublevel
of maximum $m$ of the F=3 level. However, off-resonant pumping
through the excited state $|6P_{3/2},F=3\rangle$ in the presence of large
Doppler shifts can eventually drive all the atomic population into
the $|F=4\rangle$ dark state. To prevent this depumping, an additional
$\sigma^{+}$ repump field is used on the transition
$|F=4\rangle \rightarrow |F'=4\rangle$. In this way, a significant fraction of the
atoms is maintained in $|6S_{1/2},F=3\rangle$ \cite{Cviklinski2008}.

Control and probe fields are collimated (5mm diameter) and with respective powers of 200 mW and 150 nW. They are
combined on a polarizing beam splitter and then pass through a
$\lambda/4$ to enter the cell with circular polarization.
The probe field is then extracted using a $\lambda/4$ and a
Glan polarizer with high extinction ratio ($10^6$). The repump
beam has a slightly larger beam size and its power is adjusted
during the experiment to keep the optical density constant.

As explained above, the disappearance of the EIT is predicted to
be due to some specific velocity classes. In order to exclude
these atoms from the interaction, we use one or two additional
depumping beams  detuned by $\Delta_D$ from the
$|F=3\rangle \rightarrow |F'=4\rangle$ transition for non-moving atoms. They enable to burn specific holes in the velocity distribution.
For experimental convenience, they are contra-propagative with the
control and probe fields, with an angle around 3$^{\circ}$.
In order to remove atoms that see the probe field frequency
shifted by $\Delta_{0}$, a counterpropagating depumping beam must
have a frequency detuned by $\Delta_D=\Delta_{0}$. The atoms
are then efficiently pumped into the hyperfine ground state F=4
and do not contribute anymore to the interaction.

\begin{figure*}[htpb!]
\includegraphics[width=1.5\columnwidth]{figure3}
\caption{Transparency enhancement by reshaping of the velocity distribution. (a) Transmission of the probe field as a function of the detuning $\Delta_{2ph}$. Curve
(1): without depumping beams, Curve (2): one depumping beam with
$\Delta_D$=40 MHz and Curve (3): two depumping beams with
$\Delta_D$=40 MHz and 85 MHz. The transmission is
normalized to the transmission for large detunings. The
experimental powers are 6 mW for the depumpers, 4.5 mW for the
repumper without depump, $5.5$ mW with one depumps and $7.5$ mW
with two depumps (see text). (b) Theoretical predictions
calculated from Ref. \cite{Oxy2011}. $\Omega= 2.3\gamma$,
corresponding to the experimental value, with $\gamma=2\pi\times
5.2$ MHz. We include the ground state decoherence due to the dephasing between control and probe lasers experimentally estimated to be $\gamma_{sg}=0.077\gamma$. This value corresponds to twice the linewidth of these independent lasers. The
inset gives the velocity distribution extracted from the data and
used for the model.} \label{courbe_exp_theo}\vspace{-0.3cm}
\end{figure*}

Transmission spectra are obtained by scanning the probe
detuning, which corresponds to scanning the two-photon detuning
$\Delta_{2ph}$. Figure \ref{courbe_exp_theo}(a) gives the spectrum, with and without reshaping of the atomic
velocity distribution. Curve (1) shows a very weak EIT peak near
$\Delta_{2ph}=0$ as expected in such a broadened medium with
several excited states. We then send a depumping beam.
A broad peak, corresponding to the hole burnt in the velocity
distribution, appears in the transmission spectrum (curve (2)). As can be seen, the EIT peak is significantly enhanced.
The best transparency is obtained for detunings close to the
predicted value of $\Delta_D=$ 40 MHz, confirming that
the EIT is recovered when the detrimental effect of these atoms is
suppressed. The depth of the hole created in the
distribution by the depumping beam at 40 MHz being
saturated for a power of about 6mW, we tested the effect of a
second depumping beam, detuned from the optimal frequency and
corresponding to atoms with $\Delta_D=$85 MHz. As shown by curve
(3), the transparency is slightly increased, since this beam removes some
more atoms which have non-zero absorption in the EIT window.

\section{Discussion and conclusion}
These results are in very good agreement with the transmission
curves calculated from the model developed in Ref. \cite{Oxy2011} and given in Fig. \ref{courbe_exp_theo}(b). In this case, we have
used the velocity distributions extracted from the experimental
data to compute the susceptibility of the atomic ensemble, as
shown in the inset of Fig. \ref{courbe_exp_theo}(b). Even in the
absence of depumping beams, the distribution is non
Gaussian, since it is strongly modified by the control and the
repump beams \cite{Smith,Macquire,Lindvall}. It can be seen that
the effect of one and two depumping beams on the EIT is correctly
reproduced. Because of the distorted velocity distribution, this
model also correctly predicts a small EIT feature in the absence
of depumping, in contrast to the model of Fig. \ref{schema_3D}(b).

Finally, in order to quantify the EIT enhancement, we
introduce  an EIT contrast $C$ defined as the ratio
$C=(t_{max}-t_{min})/(1-t_{min})$ with $t_{max}$ and $t_{min}$
being the probe transmittance at the maximum and on
the side of the EIT peak.  In our experiment, $C$ is increased by
a factor  5, which can be shown to yield a decrease of the group
velocity by a factor of the same order of magnitude. This shows
the potential of our method for efficiently improving
EIT-based processes. We note that the full theoretical model \cite{Oxy2011} predicts an enhancement factor around 8. The main limitation in our experimental demonstration is coming from the contrast of the hole burned in the velocity distribution. A detailed study of the hole burning dynamics is expected to bring further enhancement of the EIT feature \cite{Smith,Macquire} .

In summary, we have reported a detailed experimental
characterization of the EIT properties of a medium with Doppler
broadening and multiple excited levels. It can be shown that some
specific velocity classes are mainly causing the suppression of
transparency. We have proposed a procedure to remove these atoms
from the interaction via a well-designed reshaping of the
atomic velocity distribution. Our observations confirmed the
general mechanism and enabled to demonstrate a strong enhancement
of the transparency. This study may bring new applications and
offer significant improvements in various settings based on EIT or
related effects, including in quantum information science and
metrology. Moreover, beyond the specific medium used here, i.e
alkali-metal atoms at room temperature, our method, which allows
an efficient engineering of the EIT properties of an
inhomogeneously broadened medium, can be extended to various
atom-like physical systems presenting simultaneously large
broadening and multiple levels, e.g. in rare-earth doped crystals,
quantum dots or  nitrogen-vacancy centers in diamonds.


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

Silicon Avalanche PhotoDiodes (APDs) have become in recent years a convenient and common photo-detector choice because of the compact size, good radiation hardness, and insensitivity to intense magnetic fields.
Avalanche PhotoDiodes replaced traditional photo-multipliers tubes in many particle physics detectors. APDs have been successfully used in the CMS detector at the CERN LHC, employing more than 150k APDs in the electromagnetic calorimeter \cite{CMS-ECAL}, and in the ALICE Photon Spectrometer (PHOS), equipped with 18k sensors \cite{PHOS-ECAL}. The PANDA detector at FAIR foresees to use more than 15k Large Area APDs for the electromagnetic calorimeter readout, operating them at $-25$ $^{\circ}$C \cite{PANDA-ECAL}. Other significant examples of large-scale detectors employing APDs for light readout include the Forward Tagger Calorimeter in Hall-B at Jefferson Laboratory \cite{FT-CAL} and the HPS-Ecal Calorimeter \cite{HPS-CAL}, both currently under development.

The intrinsic gain and the dark current of  Avalanche PhotoDiodes strongly depends on the bias voltage and on the operating temperature. It is therefore necessary to characterize them before installation in any experimental setup to be able to work at the selected working point.
This is even more critical for detectors employing Avalanche PhotoDiodes for multiple channel readout, as for electromagnetic calorimeters, since differences between working points could induce degradation of the full detector performances. Mapping the gain of each sensor also permits to group together those with close working point and supply them with the same bias voltage, thus reducing the number of independent high voltage channels.

In this paper we present our realization of a new a facility for the automatic characterization of Avalanche PhotoDiodes in the $-2$ $^{\circ}$C - $25$ $^{\circ}$C temperature range. The facility was first developed to measure the 380 Large Area APDs (model Hamamatsu S8664-1010) employed in the CLAS12-Forward Tagger Calorimeter. This was the first massive usage of the system. The good performances obtained validated the design of the facility, which was used again later to characterize the 516 Large Area APDs employed in the HPS-Ecal detector.

\section{LA-APDs characterization procedure}\label{sec:ApdProcedure}

\subsection{Intrinsic gain measure}

The procedure employed in the facility to measure the APDs internal gain ($G$) at fixed temperature ($T$) as a function of the bias voltage ($V$) is as follows. The dark current $I_{off}$ and the photo-current $I_{on}$  due to a continuous illumination  are measured at different values of the reverse bias voltage with a picoammeter. The gain is derived using the following equation:

\begin{equation}\label{eq:apd1}
G(V)=\frac{I_{on}(V)-I_{off}(V)}{I_{on}(G=1)-I_{off}(G=1)} \; \; \; ,
\end{equation}
where $I_{on}(G=1)$ and $I_{off}(G=1)$ are, respectively, the values of the photo-current and the dark current when the internal gain of the APD is equal to one, i.e. when the bias voltage is sufficiently low so that the avalanche mechanism is not active. Since the procedure employed to calculate the gain involves the normalization to the unitary gain current, the only requirements for the light source are the stability during the measurement time and an intense light emission to provide a significant $I_{on} (G=1)$ current, not lower than $\simeq 10$ nA, to guarantee a good signal over noise ratio in the current measurement.
Being the spectral response of the sensor typically peaked at 420 nm, a standard blue LED can be employed.
\begin{figure}[tpb]
\begin{centering}
\includegraphics[width=0.6\textwidth]{fig1a.eps}
\caption{\label{fig:apdF1} \small Typical LA-APD photo-current $I_{on}$ (closed black markers) and dark current $I_{off}$ (open black markers) behaviour, as a function of the bias voltage $V$ (these data were obtained measuring a LA-APD, model Hamamatsu S8664-1010). The corresponding internal gain $G$ is also reported (red markers). This measurement was performed at $18.5 \pm 0.1$ $^{\circ}$C, in the voltage range between 0 and 400 V.}
\end{centering}
\end{figure}

The typical behavior of an APD photo-current $I_{on}$ and dark current $I_{off}$ is shown in Figure~\ref{fig:apdF1}. After an initial plateau, which corresponds to the unitary gain, $I_{on}$ grows rapidly for increasing bias voltage. $I_{off}$ remains smaller than $I_{on}$ by a factor $\simeq 100 - 1000$. The value of the denominator in equation (\ref{eq:apd1}) thus corresponds to the initial plateau of $I_{on}$. To reduce the measurement time, the voltage scan is performed with non-uniform steps in the above mentioned bias range, with denser measurements in the two regions of interest: the unitary-gain plateau and the neighborhood of the foreseen working point ($G=150$ for the specific example here reported).
The corresponding internal gain $G$ is also shown in Figure~\ref{fig:apdF1}. The unitary gain plateau ($G=1$) is visible for bias values lower than $\simeq 100$ V, while for bigger values $G$ increases rapidly.
The measured dark current $I_{off}$ is reported in Figure \ref{fig:apdF2}, left panel, as a function of the intrinsic gain $G$. It shows a linear behavior, as expected from the most common interpretation of a LA-APD dark current as a sum of two terms: the first (``surface current'') being independent of $G$, and the second (``bulk current'') being directly proportional to it \cite{PANDA-ECAL}.

\begin{figure}
\begin{center}
\subfigure{\includegraphics[width=0.48\textwidth]{fig2.eps}} \quad
\subfigure{\includegraphics[width=0.48\textwidth]{fig3.eps}}
\caption{\small Left: LA-APD dark current $I_{off}$ as a function of the LA-APD gain $G$. Right: LA-APD intrinsic gain variation as a function of the bias voltage. The corresponding sensor gain is also reported on the top axis.}
\label{fig:apdF2}
\end{center}
\end{figure}

The relative APD gain dependence on the bias voltage is evaluated numerically, by computing the corresponding incremental ratio:
\begin{equation}
\frac{1}{G}\frac{\partial G}{\partial V} \simeq \frac{1}{G(V)} \frac{G(V+\varepsilon)-G(V)}{\varepsilon} \; \; \; ,
\end{equation}
with $\varepsilon=0.1$ V. $G(V+\varepsilon)$ is evaluated from the measured $G(V)$ data points, employing a spline interpolation. A typical result is shown in Figure~\ref{fig:apdF2}, right panel.

\subsection{Temperature and voltage dependence}

The temperature dependence of the APD intrinsic gain and dark current is investigated by comparing measurements performed at different temperatures.
Figure \ref{fig:apdF4}, left panel, shows an example for a sensor characterized in the temperature range between $-1.5$  $^{\circ}$C and $+22.5$ $^{\circ}$C. The same data are reported in Figure \ref{fig:apdF4}, right panel, linearly extrapolated in the whole temperature range. We observed that the isogain curves in the $V-T$ plane are approximately straight lines (in the region where $G>1$, i.e. outside the unitary gain plateua). Therefore, the APD gain can be actually parametrized as a linear combination of these two variables:
\begin{equation}\label{eq:apd2}
G(V,T)=G(\alpha V - \beta T) \equiv G(x) \; \; \; ,
\end{equation}
with $\alpha$ and $\beta$ two proper coefficients to be measured experimentally, and $x=\alpha V - \beta T$.
\begin{figure}
\begin{center}
\subfigure{\includegraphics[width=0.48\textwidth]{fig4.eps}}\quad
\subfigure{\includegraphics[width=0.48\textwidth]{fig5a.eps}}
\caption{\label{fig:apdF4} \small Left: LA-APD intrinsic gain $G$ for different temperatures.
$+: -1.44$  $^{\circ}$C, $*: 2.51$ $^{\circ}$C, $\circ: 6.47$ $^{\circ}$C, $\times : 10.44$ ${\circ}$C,
$\bullet: 14.43$  $^{\circ}$C, $\blacktriangle: 18.42$  $^{\circ}$C, $\blacktriangledown: 22.39$  $^{\circ}$C.
Right: LA-APD gain (color-scale), as a function of the bias voltage $V$ and operating temperature $T$. The isogain curves are reported in black.}
\end{center}
\end{figure}

We exploited this behavior to derive the bias voltage $V_0$ corresponding to a given gain $G_0$, at temperature $T_0$, as follows.
First, the photo-sensor gain is measured at a few different temperatures $T_i$, and the bias voltage $V_i$ corresponding to a certain fixed gain $G$ is evaluated.
Then, the ratio $k\equiv\alpha/\beta$ is calculated by performing a best-fit with a linear function to the $T_i$ vs $V_i$ data points. The procedure is repeated for different gain values, to check the stability of the parameter $k$. An example is reported in Figure~\ref{fig:apdF6}, left panel. The corresponding value of $k$ is $1.26 \pm 0.01$.
$V_0$ is finally calculated as:
\begin{equation}
V_0 = V_i + k^{-1} \cdot (T_0 - T_i) \; \; \; ,
\end{equation}
where $V_i$ is the bias voltage corresponding to $G_0$ at any of the temperatures $T_i$.

The intrinsic gain relative variation with respect to the operating temperature follows from Eq.~\ref{eq:apd2}:
\begin{equation}
\left\{
\begin{array}{c}
\frac{\partial G}{\partial V} = \alpha G^{\prime}(x) \\
\frac{\partial G}{\partial T} = \beta G^{\prime}(x)
\end{array}
\right.
\Rightarrow
\,
\frac{1}{G}\frac{\partial G}{\partial T} = k^{-1} \frac{1}{G} \frac{\partial G}{\partial V}
\end{equation}

\subsection{Benchmarking}
We identified a set of quality-checks to ultimately decide if a measured sensor is properly operational or not. Any APD not fullfilling any one of these requirements has to be characterized again and, if still presenting anomalies, discarded.
\begin{itemize}
\item{The measured dark-current $I_{on}$ and the photo-current $I_{off}$ as a function of the bias voltage $V$ must be checked for all the measured temperatures. These data are required to have, qualitatively, the same behavior as reported in Figure~\ref{fig:apdF1}, i.e. the photo-current must show a plateau for $V \lesssim 50$ V, and then grow with approximately an exponential behavior. The dark-current $I_{off}$ should follow the $I_{on}$ behavior, and always be lower than it by at least one order of magnitude.}
\item{The dark-current behaviour $I_{off}$ as a function of the photo-detector gain $G$ has to be checked for all the measured temperatures. It should exibit a linear dependence, $I_{off} = c_1 + c_2 G$, and the constant contribution $c_1$ should decrease with the operating temperature.}
\item{The relative gain variation with respect to the bias voltage $\frac{1}{G}\frac{dG}{dV}$ must be finally checked, as a function of the gain $G$. Results obtained at different temperatures should approximately superimpose, for $G\gtrsim 10$, as follows directly from the gain functional dependence on bias voltage and temperature discussed before. A typical example is shown in Figure~\ref{fig:apdF6}, right panel.}
\end{itemize}

\begin{figure}[tpb]
\begin{centering}
\subfigure{\includegraphics[width=0.48\textwidth]{fig6.eps}}\quad
\subfigure{\includegraphics[width=0.48\textwidth]{fig7.eps}}
\caption{\label{fig:apdF6} \small Left: bias voltage corresponding to different LA-APD gain values, as a function of operating temperature. The curves are the results of the best fits performed with a linear function to the different data-sets to evaluate the LA-APD $k$ parameter (the lowest temperature point for $G=50$ was intentionally excluded from the fit). Right:  Measured $\frac{1}{G}\frac{dG}{dV}$ vs $G$ curves, for different operating temperatures. Same symbols as in the previous Figure.}
\end{centering}
\end{figure}

\section{Facility for APD characterization}\label{sec:Facility}

The layout of the facility for automatic APD characterization that we developed is shown in Figure~\ref{fig:Facility1}. The facility can measure up to 24 sensors at once, characterizing them in the voltage range between 0 and 500 V. The operating temperature can be varied from -2 $^{\circ}$C to +25 $^{\circ}$C. The measurement at each temperature takes approximately 2.5 hours. Therefore, one day is required to fully characterize 24 sensors from the minimum to the maximum temperature, with 8 measurements at $\simeq 3.5$ $^{\circ}$C intervals. The facility has been used to characterize the following APD models: Hamamatsu S8664-55 and Hamamatsu S8664-1010. There are no limitations to characterize other APD models, providing they have the same mechanical layout for the connection pins as the two aforementioned models.

\begin{figure}[tpb]
\centering
\subfigure{\includegraphics[height=3.5in]{apparatus1.eps}}\quad
\subfigure{\raisebox{15mm}{\includegraphics[width=0.48\textwidth]{apparatus.eps}}}
\caption{\small Left: simplified scheme of the facility for LA-APD gain measurement. Right: picture of the facility, showing the two cold copper plates with holes for LA-APDs connection on the underlying PCB cards. One LA-APD is installed on the left copper plate.
}
\label{fig:Facility1}
\end{figure}
Each installed sensor is characterized using the procedure described in the previous section. The dark current $I_{on}$ and the photo-current $I_{off}$ are measured for different bias voltages to obtain the internal gain. In order to employ a single picoammeter to measure the currents, and a single high-voltage source, a custom current multiplexer was developed. The current multiplexer is the core of the facility (see Figure~\ref{fig:apdF7}). During operation, the APD cathodes are all connected to the common HV source via independent bias resistors, while the anodes are connected to the multiplexer inputs. The multiplexer single output is connected to the picoammeter. In this way, only one APD at a time is physically connected to the instrument, while the other 23 are coupled to ground. We designed the current multiplexer using mechanical relays (type SPDT, ``single pole, double throw''\footnote{A SPDT relay is a 3-terminals device, with the common terminal connect to either of the two others, depending on the state of the magnetic coil.}), because the measured currents are of the order of 10-100~nA and other types of electronic switches, such as those based on transistors, would introduce spurious currents of the same order of magnitude, leading to incorrect gain measurements. Tyco ``T7SS5E4-12'' relays were selected because of the high isolation between the output contacts and the PCB mounting option. Relays are driven by four independent Darlington Transistor Arrays (Texas Instruments ULN2003A), that, in turn, are controlled trough a digital interface on the DAQ PC (NI 9403).

\begin{figure}
\centering
\subfigure{\includegraphics[height=3in]{circuit1.eps}}\quad
\subfigure{\raisebox{5mm}{\includegraphics[height=2.5in]{fig8.eps}}}
\caption{\small Left: Schematic of the gain measurement circuit (just three channels are reported). ``HV'' is the common APDs high voltage source, while ``A'' is the picoammeter. Right: the current multiplexer.
}
\label{fig:apdF7}
\end{figure}

The 24 APDs under measurement are mounted on two drilled copper plates, both hosting an LED in the center. Each sensor is mounted in correspondence to a hole, and connected to a PCB on the opposite side of the plate. The APDs are arranged circularly around the LED, which is also connected to the PCB. The PCB distributes the HV bias to each sensor, feeds their currents to the input of the current multiplexer, and provides current to the LED. The facility includes a cooling system to keep LA-APDs at a constant temperature. The coolant flows in a pipe connected to an external programmable chiller and welded to each of the two plates. The temperature is continuously monitored via 2 thermo-resistors mounted on each plate. The chiller provides temperature feedback loop, measuring the temperature of the return liquid, with 0.1 $^{\circ}$C stability. Furthermore, during operation, the copper assembly is inserted in a vacuum- and light-tight box flushed with nitrogen to prevent moisture formation.

The system, controlled via a Labview program running on the DAQ PC, works as follows. During the initial setup, the user enters the values for the temperature scan and the IDs of the sensors under measurement. For each programmed temperature, the APDs are individually enabled and characterized. At the end of the full cycle data are analyzed to calculate the gain of the APDs as a function of the bias voltage and temperature, the dark current dependence on the gain, and the relative gain variation at the different temperatures. The raw data and other relevant parameters, such as the bias voltage corresponding to the nominal gain at room temperature and the relative variation with respect to the temperature and the bias voltage in the neighborhood of the nominal working point, are then recorded to a file. A set of histograms is also produced, to let the user perform the quality check previously described.

\subsection{Systematic effects}
During the facility commissioning we investigated the relevance of possible systematic effects in the measurement, such as those due to long term fluctuations introduced by temperature drifts, residual moisture formation, and variation in the LED emission.
The gain of the same LA-APD has been repeatedly measured over a time period of 15 days, and the bias voltage corresponding to a fixed gain at room temperature ($G=150 \, @ \, T=+18$ $^{\circ}$C for the specific sensor) has then been calculated for each measurement. Results showed that the working point was constant within less than 0.1 V for all measurements.
This corresponds to a relative error on the measured gain approximately equal to $0.6 \% $, being $\frac{1}{G}\frac{\partial G} {\partial V} \simeq 6 \% / V $ in the neighborhood of the LA-APD selected working point ($G=150$). The same systematic effect was found on the other measured quantities, such as the dark current and the $k$ parameter.

\section{FT-Cal LA-APD characterization results}

\begin{figure}[tpb]
\centering
\includegraphics[width=0.47\textwidth]{fig9.eps}
\includegraphics[width=0.47\textwidth]{fig10.eps}
\caption{\small \label{fig:APDF9}
Left: correlation between the \textit{measured} bias voltage for $G=150$ at $+20$ $^{\circ}$C ($y$ axis) and the value reported by the manufacturer for $G=50$ at $+25$ $^{\circ}$C ($x$ axis). Right: absolute bias voltage difference for $G=150$ between the measure performed at $0$ $^{\circ}$C and the extrapolated result from $+20$ $^{\circ}$C.}
\end{figure}

The first large-scale use of the APD characterization facility was to measure the LA-APD employed in the FT-Cal detector of the CLAS12 experiment at Jefferson Laboratory. The 380 sensors were characterized using the above facility within one month. Results were analyzed to identify and select the properly working sensors. As a result, 17 samples were discarded, since they did not fulfill the previously described data-quality checks, either after a second measurement. Figure \ref{fig:APDF9}, left panel, shows the comparison between the measured bias voltage corresponding to APD gain $G=150$ at $+20$ $^{\circ}$C, and the value declared by the manufacturer for the working point $G=50$ at $+25$ $^{\circ}$C, for the accepted samples.
The good correlation demonstrates the reliability of the system.
The right panel, instead, shows the absolute difference between the bias voltage measured for $G=150$ at 0 $^{\circ}$C and the corresponding value extrapolated from the measure at  $20$ $^{\circ}$C, using the procedure previously described. The average difference of $\simeq$ 0.25 V corresponds approximately to a gain uncertainty of $\simeq 2\%$ (see Figure \ref{fig:apdF6}). This is a worst-case scenario, since in reality APDs are characterized at temperatures close to the foreseen working point, thus reducing the extrapolation error. The result, however, proves the reliability of the gain extrapolation procedure in the whole $(V,T)$ parameter space.

\section{Conclusions}

We developed a facility to automatically characterize Avalanche PhotoDiodes in the -2 $^{\circ}$C - 25 $^{\circ}$C temperature range. Such a facility can characterize up to 24 sensors at once, within one day of operation, measuring APDs from the minimum to the maximum temperature in $\simeq 3.5$ $^{\circ}$C intervals . For each sensor, the bias voltages corresponding to the nominal gain of 150 at different temperatures, and the relative gain variation with respect to the voltage and temperature are recorded.
Particular care was taken to check possible systematics on the measured gain, that have been evaluated to be of the order of $0.6\%$, for a sensor gain $G=150$.

\section{Acknowledgments}

The authors want to acknowledge the electronic group of INFN-Genova section for the support given during the design of this facility and A. Balbi for help during the construction.

redpajama_set_name: RedPajamaArXiv


text:
\section{I. Introduction}

Although supersymmetry has so far eluded discovery at the LHC \cite{ATLAS20,CMS20}, it remains an important component for building models
beyond the Standard Model, particularly when going to grand unified scales and beyond. It is well known that supersymmetry can provide for the naturalness of the weak scale \cite{Maiani:1979cx}, the unification of gauge couplings \cite{Ellis:1990zq}, and a cold dark matter candidate \cite{gold,ehnos}.
As supergravity~\cite{cremmer,susy} is the extension of supersymmetry which includes gravity, it is the
natural framework for any cosmological scenario involving supersymmetry.
Thus, the gravitino also becomes a component in post-inflationary cosmology.

The gravitino has long since been a cosmological headache \cite{prob,sw,eln,ego}.
It has the potential to overclose the universe if it is stable
(and hence a dark matter candidate), or lead to an overabundance of the lightest supersymmetric particle if it decays. While inflation may dilute any primordial gravitinos \cite{eln}, they are reproduced during reheating after inflation \cite{nos,ehnos,kl,ekn,oss,Juszkiewicz:gg,mmy,Kawasaki:1994af,Moroi:1995fs,enor,Giudice:1999am,bbb,cefo,kmy,stef,Pradler:2006qh,ps2,rs,kkmy,egnop,Garcia:2017tuj,Eberl:2020fml}. Decaying gravitinos may also upset the concordance of big bang nucleosynthesis \cite{foyy} and these constraints result in limits on the gravitino abundance which ultimately translate into limits on the reheat temperature \cite{bbn,cefo,ceflos175,Kawasaki:1994af,stef}.

In addition, there is also a non-thermal contribution to the gravitino abundance stemming from the non-perturbative decay of the inflaton when supersymmetry is broken during inflation \cite{Kallosh:1999jj,Giudice:1999yt,Giudice:1999am,Kallosh:2000ve,Nilles:2001ry,Nilles:2001fg,Ema:2016oxl}. However, it was shown that what is produced in this process is the goldstino, that is, the combination of fermions contributing to supersymmetry breaking at the time of their production.
At the inflationary scale, the goldstino is typically dominated by the inflatino, the fermionic partner of the inflaton. The longitudinal component of the gravitino at low energy is associated with the true goldstino which may be unrelated to the inflatino produced in reheating \cite{Nilles:2001ry,Nilles:2001fg}.
The production of inflatinos may be problematic, particularly if the mass of the inflatino is less than the mass of the inflaton, but may be suppressed and the final gravitino abundance is model dependent \cite{Nilles:2001my}.

Recently, it has been claimed that if the sound speed, $c_s$, for gravitinos
were to vanish, there would be a catastrophic non-thermal production of slow gravitinos \cite{Kolb:2021xfn,Kolb:2021nob}.
This is due to the lack of suppression in the production of high momentum modes in the helicity 1/2 spectrum.
While $c_s$ may indeed vanish in certain models of inflation derived from supergravity, as in the case of the non-thermal production of the longitudinal mode, mixing with other fermionic fields whose scalar partners have non-vanishing time derivatives precludes the runaway production of gravitinos.
Nevertheless, in some realizations of non-linear supergravity with constrained fields, the inflatino may not be present in the spectrum, and indeed catastrophic production may occur.

In this paper, we consider the conditions which lead to $c_s = 0$. We look at several
models of inflation in supergravity theories
and distinguish those for which $c_s = 0$ at some point in the post-inflationary oscillation period of the inflaton. We stress that $c_s = 0$ is not a generic feature of supergravity inflation models. Further we argue that even if $c_s = 0$, in the absence of constrained fields, catastrophic gravitino production does {\em not} occur. Indeed even in constrained models with a nilpotent field (eliminating a scalar from the spectrum) \cite{Rocek:1978nb,Komargodski:2009rz,Antoniadis:2014oya,Ferrara:2014kva,Kallosh:2014via,DallAgata:2014qsj,Dudas:2015eha,Ferrara:2015gta,Ferrara:2015cwa}, there is no catastrophic production.  Only when a second, orthogonal constraint \cite{Ferrara:2015tyn,Carrasco:2015iij,DallAgata:2015zxp,Kallosh:2016hcm,DallAgata:2016syy} which eliminates, the pseudoscalar, fermionic and auxiliary partners of the inflaton, can catastrophic production occur when $c_s = 0$. In \cite{hase}, only models with both constraints were considered leading to a truncated equation of motion for the longitudinal component of the gravitino \cite{Ferrara:2015tyn} and explosive particle production.  While the nilpotent constraint can easily be viewed as the infrared limit of a heavy sgoldstino (the scalar field associated with supersymmetry breaking), the orthogonal constraint appears to require higher derivative operators in the action \cite{DallAgata:2016syy}. Indeed we would argue that the constraints from gravitino production signal a further problem for the orthogonal constraint rather than imposing conditions on general models of inflation in supergravity.

In what follows, we first consider in Section \ref{sec:theta} the calculation of the sound speed in supergravity theories. It is easy to see that in models with a single chiral field, $c_s = 1$. We next consider in Section \ref{sec:slow} several models of inflation with supersymmetry breaking showing that $c_s$ is quite model dependent.
In some cases $c_s = 0$ occurs at some specific points during the oscillation of the inflaton, while in other often studied models, $c_s \simeq 1$ always. In Section \ref{sec:theta-Y}, we reconsider the equations of motion for the longitudinal component of the gravitino. We show clearly that even when $c_s = 0$, there is no enhanced production of gravitinos. The nilpotency conditions are discussed in Section \ref{sec:nil}. With a single nilpotent field, our conclusions are left unchanged.  Only when a second orthogonal condition is imposed, and the inflatino is eliminated from the spectrum, is there the potential for explosive gravitino production.  A discussion and our conclusions are given in Section \ref{sec:conclusions}.

\section{Gravitino sound speed in supergravity}
\label{sec:theta}

It is argued in \cite{Kolb:2021xfn,Kolb:2021nob} that the vanishing of the gravitino sound speed is accompanied by a catastrophic gravitino overproduction during the first set of  inflaton oscillations. The mathematical reason for this statement can be understood from the linearized theory of an uncoupled longitudinal gravitino. The gravitino has a dispersion relation of the type $\omega_k^2=c_s^2 k^2 + a^2 m^2(t)$, $a$ being the scale factor and $k$ the comoving momentum. As is well known from studies of preheating \cite{preheating,stb, kt,Greene:1997fu}, a non-adiabatic change of the frequency, $\omega_k' / \omega_k^2 \gg 1$ (where prime denotes derivative with respect to conformal time), results in significant particle production. Normally, the $k^2$ term ensures that the variation is adiabatic in the deep UV, so that modes of high momenta remain in their vacuum state. This is not the case if $c_s^2 =0$. In fact, the production obtained in \cite{hase,Kolb:2021xfn,Kolb:2021nob} is formally divergent in the UV, and it is presumably regulated by higher-order operators. This would in any case result in the production of an unacceptably large gravitino abundance.

As we shall see in Section \ref{sec:theta-Y}, this problem is typically not present in multi-field models, since in this case the longitudinal gravitino is coupled already at the linearized level to a linear combination $\Upsilon$ of the fermions in the theory \cite{Kallosh:2000ve,Nilles:2001fg}. The gravitino momentum term is then replaced by a matrix in field space, that is non-singular when $c_s^2 = 0$. Therefore, the physical eigenvalues of the system do not vanish, and the catastrophic production in the UV is avoided.

We begin by first computing $c_s$ in supergravity.
The equations for the transverse and longitudinal gravitino component presented in this work follow the formalism of Ref.~\cite{Kallosh:2000ve}. In that work, the FLRW line element is chosen according to $ds^2 = a^2 \left( \tau \right) \left[ - d \tau^2 + d \vec{x}^2 \right]$, where $\tau$ is the conformal time. We denote by $\gamma^\mu$ the gamma matrices in flat spacetime. We work in units of $M_p =1$, where $M_p$ is the reduced Planck mass.

The equation of motion for the transverse gravitino component is
\begin{eqnarray}
\left( \gamma^0 \partial_0 + i \gamma^i k_i + a m_{3/2} \right) \psi^T = 0
\label{eq-psi32}
\end{eqnarray}
where $\vec{k}$ is the comoving momentum, and $\partial_0$ is the derivative with respect to the conformal time. The propagation of the
transverse mode of the gravitino
is not directly affected by the supersymmetry breaking mechanism, and obeys a classical Dirac equation with a speed of sound $c_s=1$. The gravitino mass is given by
\begin{equation}
m_{3/2} = {\rm e}^{K/2} \, \left\vert W \right\vert \;,
\end{equation}
where $K$ and $W$ denote the K\"ahler potential and the superpotential, respectively.

The equation for the longitudinal gravitino will be presented and discussed in Section \ref{sec:theta-Y}. Here we discuss one important difference with Eq.~(\ref{eq-psi32}) for the longitudinal component. Namely, the momentum-dependent term in the equation for the longitudinal component is multiplied by a function of the background scalar fields, that, once squared, is identified as the square of the longitudinal gravitino sound speed $c_s^2$,
\begin{equation}
c_s^2 = \frac{\left(p-3 m_{3/2}^{2}\right)^{2}}{\left(\rho+3 m_{3/2}^{2} \right)^{2}}+\frac{4 \dot{m}_{3/2}^2}{\left(\rho+3 m_{3/2}^{2} \right)^{2}} \, ,
\label{Eq:generic32}
\end{equation}
where $p$ is the pressure, $\rho$ is the energy density, and dot denotes the derivative with respect to cosmological time.
Ref.~\cite{Kolb:2021xfn} provides a rather compact expression for this quantity in a supergravity model, namely
\begin{equation}
c_s^2 = 1 - \frac{4}{
\left(\vert\dot{\varphi}\vert^2 + \vert F \vert^2 \right)^2} \,
\left\{
 |\dot{\varphi}|^2  |F|^2  -
\left\vert \dot{\varphi}  \cdot F^*  \right\vert^2 \right\}  \;,
\label{vs2}
\end{equation}
where $\varphi$ is the multiplet of scalar fields in the model, and the $F$-term is
\begin{equation}
F^i \equiv {\rm e}^{K/2} K^{ij^*} \, D_{j^*} W^* \;,
\label{DWF}
\end{equation}
where, using standard supergravity notation, $K^{ij^*}$ is the inverse of the K\"ahler metric
\begin{equation}
K_{ij^*} \equiv \frac{\partial^2 K}{\partial \varphi^i \, \partial \varphi^{j*}} \;,
\label{Kij}
\end{equation}
while
\begin{equation}
D_i W \equiv \frac{\partial W}{\partial \varphi^i} +  \frac{\partial K}{\partial \varphi^i} \, W \,.
\end{equation}
The dot operator in Eq.~(\ref{vs2}) denotes a  scalar product with the K\"ahler metric (\ref{Kij}), namely $\vert \dot{\varphi} \vert^2 = \dot{\varphi}^i \, K_{ij^*} \, \dot{\varphi}^{j*}$, and analogously for the other terms.
Notice that due to the Cauchy-Schwarz type inequality
$|\dot{\varphi}|^2  |F|^2  \geq
\left\vert \dot{\varphi}  \cdot F^*  \right\vert^2$, causality $c_s \leq 1$ is always guaranteed to hold.

In the case of a single chiral superfield, the two terms in the curly bracket in Eq. (\ref{vs2}) are equal to each other, and therefore $c_s=1$.
Thus, $c_s^2 \simeq 1$ is expected whenever a single field dominates the kinetic energy and supersymmetry breaking in the model.

Ref. \cite{Kolb:2021xfn} considered the case in which multiple fields are relevant, and they conspire to give a vanishing or nearly vanishing $c_s^2$.
This can be achieved if
\begin{equation}
\dot{\varphi}  \cdot F^* = 0 \;\;{\rm and}\;\;
\dot{\varphi}  \cdot \dot{\varphi}^* =
F \cdot F^* \;\; \Rightarrow \;\;
c_s^2 = 0 \;.
\label{vs0}
\end{equation}
Note that the first of these conditions, $\dot{\varphi}  \cdot F^* = 0$, is realized whenever the gravitino mass is constant.
These conditions can be satisfied during inflaton oscillations after inflation. Typically, the inflaton dominates the kinetic energy, so the condition $\dot{\varphi}  \cdot F^* \simeq 0 $ generically requires that the $F$-term associated with the inflaton is small. Barring cancellations, this would typically require that both $W$ and $\frac{\partial W}{\partial \phi}$, where $\phi$ denotes the inflaton field, are small. We note that the potential energy is given by
\begin{equation}
V = F \cdot F^* - 3 \, {\rm e}^K \, \left\vert W \right\vert^2 \;,
\end{equation}
which we can approximate by $V \simeq F \cdot F^*$ if $W$ is small. Then the second condition in (\ref{vs0}) simply demands that the kinetic and the potential energy are equal to each other,  which happens twice per period of the inflaton oscillations. In Section \ref{sec:slow},  we discuss several supergravity models of inflation where these conditions are and are not achieved.

\section{Gravitino sound speed in specific models}
\label{sec:slow}

In this section we consider several specific supergravity models where $c_s^2$ is very small, or is of order one, to emphasize what aspects of the model can lead to a slow gravitino.

We start our discussion from a model constructed in \cite{Kolb:2021xfn}, where $\Phi$ is the inflaton superfield while $S$ is a superfield responsible for supersymmetry breaking. Ref.  \cite{Kolb:2021xfn} imposes that this field is nilpotent, $S^2=0$. To study the relevance of this assumption we instead use a strong stabilization mechanism for $S$ \cite{ekn3,strongpol,dlmmo,ADinf,ego,eno7,egno4,Ellis:2020xmk} (see also Section \ref{sec:nil} below):
\begin{eqnarray}
K &=& - \frac{\left( \Phi - {\bar \Phi} \right)^2}{2} + \frac{S \, {\bar S}}{ 1 + \frac{m^2}{M^2} \left\vert \Phi \right\vert^2}
- \frac{\left( S \, {\bar S} \right)^2}{\Lambda^2} \;,  \nonumber\\
W &=& M \, S + W_0 \;.
\label{kolb-Lambda}
\end{eqnarray}

The resulting potential is extremized along the real directions $\Phi = {\bar \Phi} = \frac{\phi}{\sqrt{2}} \;,\; S = {\bar S} = \frac{s}{\sqrt{2}}$. The minimum of the potential with respect to $s$ is $\phi-$dependent and given by:
\begin{equation}
\left\langle s \right\rangle_\phi = \frac{\Lambda^2}{\sqrt{6} \left( \frac{m^2 \phi^2}{2 M^2}+1\right)^2} + \mathcal{O} \left( \Lambda^4 \right) \;.
\label{smin-Kolb}
\end{equation}
We see that, as is typical for strong stabilization, $\langle s \rangle_\phi$ vanishes in the limit $\Lambda \rightarrow 0$. Inserting this into the potential, leads to %
\begin{equation}
V = \frac{m^2 \phi^2}{2} + \frac{M^2}{3} \left( 1 - \frac{1}{\left(\frac{m^2 \phi^2}{2 M^2}+1\right)^2} \right) \Lambda^2 + \mathcal{O} \left( \Lambda^4 \right) \;.
\label{Vmin-Kolb}
\end{equation}
In both Eqs.~(\ref{smin-Kolb}) and (\ref{Vmin-Kolb}), the parameter $W_0$ has been set to $W_0 = \frac{M}{\sqrt{3}} \left( 1 - \frac{\Lambda^2}{6} + \mathcal{O} \left( \Lambda^4 \right) \right)$, so to have a vanishing potential in the minimum at $\phi =0$. In the minimum, the gravitino mass is given by
\begin{equation}
m_{3/2} \Big\vert_{\phi = 0} = \frac{M}{\sqrt{3}} + \mathcal{O} \left( \Lambda^2 \right) \;.
\label{m32-min-Kolb}
\end{equation}
From Eqs.~(\ref{Vmin-Kolb}) and (\ref{m32-min-Kolb}) we see that $m$ corresponds to the inflaton mass, while (assuming gravity mediation), $M$ to the supersymmetry breaking scale in the model. Therefore, the ratio $(m/M)^2 \sim (m/m_{3/2})^2$
appearing in the K\"ahler potential is typically much greater than 1 in models with weak-scale supersymmetry breaking.

Evaluating the quantities entering in Eq.~(\ref{vs2}) leads to
\begin{eqnarray}
\dot{\varphi}^i &\simeq& \frac{\dot{\phi}}{\sqrt{2}} \left\{ 1 ,\,
-  \frac{8 \sqrt{2} \Lambda^2 M^4}{\sqrt{3} m^4 \phi^5}  \right\} \;,
\nonumber\\
F^i &\simeq& \left\{ \sqrt{\frac{2}{3}} \, \frac{2 M^5 \Lambda^2}{m^4 \phi^5} ,\, \frac{m^2 \phi^2}{2 M} \right\} \;,
\label{dF-Kolb}
\end{eqnarray}
where $i=1,2$ corresponds to the field $\Phi,\, S$, respectively. These expressions hold for $\Lambda \ll 1$ and $M^2 \ll m^2 \phi^2$, and only the dominant term has been retained in each entry.

From Eq.~(\ref{dF-Kolb}) we see that the model approximately meets the conditions (\ref{vs0}) for a vanishing sound speed. The scale of the superpotential, $M$ is much smaller than the inflationary scale, $m$, and $dW/d\phi = 0$. In particular, the inflaton provides a negligible contribution to supersymmetry breaking.
Applying Eq.~(\ref{vs2}), we find
\begin{equation}
c_s^2 = \left( \frac{\frac{\dot{\phi}^2}{2} -\frac{m^2 \phi^2}{2} - M^2}{\frac{\dot{\phi}^2}{2} +\frac{m^2 \phi^2}{2} + M^2} \right)^2 + \mathcal{O} \left( \Lambda^2 \right) \;,
\end{equation}
which indeed vanishes to $\mathcal{O} \left( \Lambda^0 \right)$ when the inflaton kinetic and potential energy are equal to each other (disregarding the subdominant $M^2$ correction).
Expanding the various terms to higher order in $\Lambda$ we find that, when the $\mathcal{O} \left( \Lambda^0 \right)$ vanishes, the minimum sound speed is
\begin{equation}
c_{s,{\rm min}}^2 = \frac{256 \, \Lambda^4}{3 \phi^2} \, \left( \frac{M}{m \phi} \right)^{10} \left[ 1 + \mathcal{O} \left( \frac{M^2}{m^2 \phi^2} \right) \right] + \mathcal{O} \left( \Lambda^6 \right) \;.
\label{vsmin-Kolb}
\end{equation}

The strong stabilization condition $\Lambda \ll 1$ provided us with an expansion parameter to organize the study of the model and present a simple analytical result. However, we see that it (or the nilpotency condition) is not origin for the near vanishing of $c_s$, which is instead mostly due to the absence of $\phi$ from the superpotential and from the smallness of the mass scale $M$.

To emphasize this, let us consider a different model with strong stabilization, which does not lead to a small gravitino sound speed. The model  \cite{Dudas:2016eej} is  also characterized characterized by the two superfields $\Phi$ and $S$, and by
\begin{eqnarray}
K &=&  - \frac{\left( \Phi - {\bar \Phi} \right)^2}{2} + \left\vert S \right\vert^2
- \frac{\left\vert S \right\vert^4}{\Lambda^2} \;, \nonumber\\
W &=& f \left( \Phi \right) \left( 1 + \delta \, S \right)
\;\;,\;\; f \left( \Phi \right) = f_0 + \frac{m}{2} \, \Phi^2  \;.
\label{mod-Emilian}
\end{eqnarray}
The K\"ahler potential in this model is relatively simple, and the origin of the inflationary potential resides in the superpotential.
Supersymmetry breaking in the vacuum is due to the constant $f_0$, though supersymmetry breaking gets a contribution from the inflaton during inflation and in its subsequent oscillations.
We can anticipate that the condition (\ref{vs0}), and in particular,
$\dot{\varphi}\cdot F^* = 0$ will not be satisfied.

As in the previous example considered, this model has real solutions. The field $s$ is stabilized to a $\phi-$ dependent $\mathcal{O} \left( \Lambda^2 \right)$ value, while the parameter $\delta$ can be set to $\delta = \sqrt{3} + \frac{\Lambda^2}{2 \sqrt{3}} + \mathcal{O} \left( \Lambda^4 \right)$ to ensure that the vacuum energy vanishes in the minimum at $\phi =0$. We then find that the potential and the gravitino mass are given by
\begin{eqnarray}
V \left( \phi \right) &=& V^{(0)} \left( \phi \right) + \mathcal{O} \left( \Lambda^2 \right) \;, \nonumber \\
m_{3/2} \left( \phi \right) & = & m_{3/2}^{(0)} \left( \phi \right)  + \mathcal{O} \left( \Lambda^2 \right) \;,
\end{eqnarray}
with
\begin{equation}
V^{(0)} \left( \phi \right) =
\left\vert f' \left( \phi \right) \right\vert^2 = \frac{m^2 \phi^2}{2} \;,
\end{equation}
and
\begin{equation}
m_{3/2}^{(0)} \left( \phi \right) = \left\vert f \left( \phi \right) \right\vert  =
f_0 + \frac{m \, \phi^2}{4}  \;,
\end{equation}
and we see that $m$ is the inflaton mass, while $f_0$ is the gravitino mass in the vacuum. In terms of the zeroth order potential and gravitino mass, the $\phi-$dependent minimum value of $s$ is given by the compact expression
\begin{equation}
\left\langle s \right\rangle_\phi = \frac{2 m_{3/2}^{(0)2} - V^{(0)}}{m_{3/2}^{(0)2}} \, \frac{\Lambda^2}{2 \, \sqrt{6}} + \mathcal{O} \left( \Lambda^4 \right) \,.
\end{equation}

For this model, we then have
\begin{eqnarray}
\dot{\varphi}^i &=& \left\{ \frac{\dot{\phi}}{\sqrt{2}} ,\, 0 \right\} + \mathcal{O} \left( \Lambda^2 \right)  \;, \nonumber\\
F^i &=& \left\{ \frac{m \phi}{\sqrt{2}} ,\,
\sqrt{3} \left( f_0 + \frac{m \, \phi^2}{4} \right) \right\}  + \mathcal{O} \left( \Lambda^2 \right) \;.
\label{dF-Emilian}
\end{eqnarray}
We see that $F^\phi$ is not suppressed, and hence we do not expect a small sound speed. Indeed, we obtain
\begin{eqnarray}
c_s^2 &=& 1 - \frac{6 \, \dot{\phi}^2 \, m_{3/2}^{(0)2}}{\left[
\frac{1}{2} \dot{\phi}^2 + V^{(0)} + 3 \, m_{3/2}^{(0)2}   \right]^2} + \mathcal{O} \left( \Lambda^2 \right) \nonumber\\
&=&
\frac{\left[\phi^2 \left( 1 + \frac{3}{8} \, \phi^2 \right) +  \frac{\dot{\phi}^2}{m^2} \right]^2 - \frac{3}{2} \frac{\dot{\phi}^2}{m^2} \phi^4}{\left[ \phi^2 \left( 1 + \frac{3}{8} \, \phi^2 \right) +  \frac{\dot{\phi}^2}{m^2} \right]^2} \nonumber\\
& + & \mathcal{O} \left( \frac{f_0}{m} ,\, \Lambda^2 \right) \;,
\end{eqnarray}
which is always of $\mathcal{O} \left( 1 \right)$.

The comparison of the two models (\ref{kolb-Lambda}) and (\ref{mod-Emilian}) shows that, as already remarked, the main cause for the suppressed sound speed is the smallness of the inflaton $F-$term. In the model (\ref{kolb-Lambda}) this happens because of the rather peculiar fact that $\Phi$ is absent from the superpotential, particularly when combined with the lack of mobility of $S$ ($S=0$ in the nilpotent limit $\Lambda \rightarrow 0$). That is, $\dot{\varphi}$ and $F^*$ are nearly orthogonal ($\dot \varphi^i\sim\{\dot \phi,0 \}$, $F^i \sim \{0,F_S \}$). In the model of (\ref{mod-Emilian}), that is clearly not the case, as there is a large contribution to supersymmetry breaking from $\Phi$ as long
as $\Phi$ is displaced from its minimum at $\Phi = 0$.
Thus, with this latter simple example, we see that
$c_s \rightarrow 0$ is not a general consequence for supergravity models of inflation, nor for the restricted set of examples with a nilpotent field.

The above two examples, lead to chaotic inflation models of the type $V = m^2 \phi^2$ \cite{Linde:1983gd}. As this model is disfavored due to the upper limit on the tensor-to-scalar ratio (the model predicts $r = 0.13$, whereas the experimental upper limit is $r < 0.06$ \cite{planck18,rlimit}), we next consider an alternative set of models leading to Starobinsky-like inflation \cite{Staro}.  These models are based on no-scale supergravity \cite{no-scale,LN} which can be derived from string models as their effective low-energy theories \cite{Witten}.
Further, there is strong relation between no-scale supergravity and higher derivative theories of gravity \cite{Cecotti,DLT,eno9}.
Starobinsky inflation models in no-scale supergravity generally require two chiral superfields, one of which is associated with the volume modulus of string theory. Depending on a choice of basis, this or the second field may serve as the inflaton \cite{eno7,enov1,building}. In the former, it is necessary to include a third chiral multiplet for supersymmetry breaking.  In the remainder of this section, we consider an example of each type.

In the first Starobinsky-like model we consider, the volume modulus serves as the inflaton \cite{FeKR,eno7}. As noted above, to include supersymmetry breaking, we need to add a third chiral superfield
\cite{egno4,Dudas:2017kfz,Ellis:2020xmk} which we denote as $Z$ and plays the role of a Polonyi field \cite{pol}.\footnote{The model described here is the  untwisted Polonyi field model of ref. \cite{egno4,Dudas:2017kfz,Ellis:2020xmk}. Analogous result are obtained for the twisted Polonyi field model of that work. To keep a notation as close as possible to that of the models (\ref{kolb-Lambda}) and (\ref{mod-Emilian}) discussed above, we have relabeled as $\Phi$ and as $S$ the inflaton and the stabilizer field that were denoted, respectively, as $T$ and $\phi$ in those papers.} The K\"ahler potential and superpotential can be written as
\begin{eqnarray}
K &=& - 3 \, \ln \left( \Phi + {\bar \Phi} - \frac{1}{3} \left\vert S \right\vert^2 + \frac{\left\vert S  \right\vert^4}{\Lambda_S^2} - \frac{1}{3} \left\vert Z \right\vert^2 + \frac{\left\vert Z  \right\vert^4}{\Lambda^2} \right) \;, \nonumber\\
W &=& m \left[ \sqrt{3} \, S \left( \Phi - \frac{1}{2} \right) + \delta \left( Z + b \right) \right] \; . \label{staro1}
\end{eqnarray}
As we will see, Starobinsky inflation is obtained for $S$ and $Z$ near 0.

As in the previous cases, the model admits real solutions, and we parametrize as $\Phi = {\bar \Phi} = \phi ,\, S = {\bar S} = s ,\, Z = {\bar Z} = z $. The potential is minimized by $\left\langle \phi \right\rangle = \frac{1}{2} + \frac{\delta^2}{3} + \mathcal{O} \left( \delta^4 ,\, \Lambda^2 \right)$, by $\left\langle s \right\rangle = \delta + \mathcal{O} \left( \delta^3 ,\, \Lambda^2 \right)$, and by $\left\langle z \right\rangle = \frac{\Lambda^2}{6 \sqrt{3}} + \mathcal{O} \left( \delta^2 \Lambda^2 ,\, \Lambda^4 \right)$, where the coefficient $b$ has been set to $b = \frac{1}{\sqrt{3}} \left( 1 - \frac{\delta^2}{6} \right) + \mathcal{O} \left( \delta^4 ,\, \Lambda^2 \right)$ so to have a vanishing potential in the minimum. In the minimum the gravitino mass is
\begin{equation}
m_{3/2} = \frac{m \, \delta}{\sqrt{3}} + m \times \mathcal{O} \left( \delta^3 ,\, \Lambda^2 \right) \;.
\end{equation}
The extremization of the potential with respect to $s$ and $z$ when $\phi$ is away from the minimum gives instead
\begin{eqnarray}
\left\langle s \right\rangle_\phi &=& \frac{4 \Lambda_S^2 \phi}{36 \left( 1 - 2 \phi \right)^2 \phi + \Lambda_S^2 \left( 1 + 4 \phi - 4 \phi^2 \right)} \, \delta + \mathcal{O} \left( \delta^3 ,\, \Lambda^2 \right) \;, \nonumber\\
\left\langle z \right\rangle_\phi &=& \frac{2 \Lambda^2 \Lambda_S^2 \phi}{3 \sqrt{3} \left[ 36 \left( 1 - 2 \phi \right)^2 \phi + \Lambda_S^2 \left( 1 + 4 \phi - 4 \phi^2 \right) \right]}  \nonumber\\
&&\quad\quad + \mathcal{O} \left( \delta^2 \Lambda^2 ,\, \Lambda^4 \right) \;.
\end{eqnarray}
The potential along this direction is
\begin{equation}
V = \frac{3m^2}{16} \, \frac{\left( 1 - 2 \phi \right)^2}{\phi^2}
+ m^2 \times \mathcal{O} \left( \delta^2 ,\, \Lambda^2 \right) \;.
\end{equation}
Note that unlike the previous examples discussed, $\phi$
does not have a properly normalized kinetic term.
For small $s$ and $z$, the canonical field is given by
\begin{equation}
\label{canrho}
\rho = \sqrt{\frac32} \ln \left( 2 \phi \right) \, ,
\end{equation}
so that the potential becomes
\begin{equation}
V = \frac{3 m^2}{4} \left(1-e^{-\sqrt{2/3}\rho} \right)^2 \, .
\label{staropot}
\end{equation}
The inflaton mass is $m$ and hence we see that $\delta \ll 1$ is the ratio of the gravitino mass to the inflaton mass and justifies the  limit that we have adopted in our analysis.
Along this direction, labeling as $1,2,3$ the fields $\Phi ,\, S ,\, Z$, respectively, we find
\begin{eqnarray}
\dot{\varphi}^i &=& \left\{ \dot{\phi} ,\, m \times \mathcal{O} \left( \delta ,\, \Lambda^2 \right) ,\, m \times \mathcal{O} \left( \Lambda^2 \right) \right\} \;, \nonumber\\
F^i &=& m \Bigg\{ \mathcal{O} \left( \delta ,\, \Lambda^2 \right) ,\,- \frac{\sqrt{3} \left( 1 - 2 \phi \right)}{\sqrt{2 \phi}} + \mathcal{O} \left( \delta^2 ,\, \Lambda^2 \right) ,\, \nonumber\\
&& \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
\mathcal{O} \left( \delta ,\, \Lambda^2 \right) \Bigg\} \;.
\end{eqnarray}
We see that $\dot{\varphi} \cdot F$ is suppressed, leading to the possibility of a small gravitino sound speed. We indeed find
\begin{equation}
c_s^2 = \left( \frac{4 \dot{\phi}^2-m^2\left(1-2 \, \phi \right)^2}{4 \dot{\phi}^2+m^2\left(1-2 \, \phi \right)^2} \right)^2 + \mathcal{O} \left( \delta^2 ,\, \Lambda^2 \right) \;,
\end{equation}
and the leading term can vanish during the oscillations of the inflaton.

We conclude this section with a second no-scale supergravity model, based on a simple Wess-Zumino form for the inflaton superpotential \cite{eno6}. In this case, the volume modulus is strongly stabilized \cite{eno7}. This simple model further emphasizes that the suppression of the sound speed is by no means a generic feature of supergravity models. The model does not require an additional field for supersymmetry breaking and can be written as \cite{egno4,Ellis:2020xmk}
\begin{eqnarray}
K \!\!&=&\!\! - 3 \, \ln \left( S + {\bar S} + \frac{\left( S + {\bar S} - 1 \right)^4}{\Lambda^2} + \frac{d \left( S - {\bar S} \right)^4}{\Lambda^2} - \frac{\left\vert \Phi \right\vert^2}{3} \right) \;, \nonumber\\
W \!\!&=&\!\! m \, \left(\frac{\Phi^2}{2} - \frac{\Phi^3}{3 \sqrt{3}}\right) + \lambda \, m^3 \;,
\end{eqnarray}
where $d$ is a constant of $\mathcal{O}(1)$, and $\lambda$ sets the scale of supersymmetry breaking.

The model admits real solutions that we parametrize as in the previous models, $\Phi = {\bar \Phi} = \phi ,\, S = {\bar S} = s$. In the minimum, $\left\langle \phi \right\rangle = 0$ and $\left\langle s \right\rangle = \frac{1}{2}$. For these values the potential vanishes, while the gravitino mass is $m_{3/2} = \lambda \, m^3$.

Extremizing the potential with respect to $s$, with the inflaton away from the minimum, we find
\begin{equation}
\left\langle s \right\rangle_\phi = \frac{1}{2} +
\frac{\phi^2}{2 \left( 6 \lambda m^2 + \phi^2 \right)^2} \, \frac{\sqrt{3}-\phi}{\sqrt{3}+\phi} \Lambda^2 + \mathcal{O} \left( \Lambda^4 \right) \;,
\end{equation}
leading to the potential
\begin{equation}
V = \frac{3 m^2 \phi^2 }{\left( \sqrt{3}+\phi\right)^2}  + \mathcal{O} \left( \Lambda^2 \right) \;.
\end{equation}
As in the previous no-scale example,
$\phi$ is not canonical and writing
\begin{equation}
\phi = \sqrt{3} \tanh \left( \frac{\rho}{\sqrt{6}} \right)
\end{equation}
leads to the same potential given in Eq.~(\ref{staropot}).

We then find, using the same notation as for the previous models,
\begin{eqnarray}
\dot{\varphi}^i &=& \left\{ \dot{\phi} ,\, m \times \mathcal{O} \left( \Lambda^2 \right) \right\} \;, \nonumber\\
F^i &=& \left\{
m \phi \, \sqrt{\frac{\sqrt{3}-\phi}{\sqrt{3}+\phi}} ,\,
- \frac{m \left( 3 \lambda m^2 + \phi^2 \right)}{2 \sqrt{3} \sqrt{3-\phi^2}} \right\} \nonumber \\
&& \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad + m \times \mathcal{O} \left( \Lambda^2 \right) \; .
\end{eqnarray}
We see that $\dot{\varphi} \cdot F$ is not suppressed, and we do not expect a suppression in the gravitino sound speed. A direct inspection of the $\mathcal{O} \left( \Lambda^0 \right)$ term indicates that this is indeed the case, though the full expression is not particularly illuminating. In the $\sqrt{\lambda} m \ll \phi ,\, \frac{\dot{\phi}}{m} \ll 1$ regime we obtain
\begin{equation}
c_s^2 \simeq 1 - \frac{m^2 \phi^4 \dot{\phi}^2}{6 \left( m^2 \phi^2 + \dot{\phi}^2 \right)^2} \;,\;\; \Lambda \ll 1 \;,\; \sqrt{\lambda} m \ll \phi ,\, \frac{\dot{\phi}}{m} \ll 1 \;.
\end{equation}

\section{Complete linearized equation for the longitudinal gravitino}
\label{sec:theta-Y}

So far, we have seen that while it is possible to find models with $c_s \simeq 0$, it is by no means a generic feature of supergravity inflation models. Next, we point out that even
if $c_s = 0$, one can not conclude that there is any enhanced production of gravitinos.
We show that this is the case by considering the equation of motion for the longitudinal gravitino, as obtained in \cite{Kallosh:2000ve} and then studied in \cite{Nilles:2001fg}.

In the unitary gauge, the dynamical variable encoding the longitudinal gravitino is the combination $\theta = \gamma^i \psi_i$, where $\psi$ is the gravitino field. In a nontrivial background, the longitudinal gravitino is coupled to another fermionic combination $\Upsilon$ at the linearized level. In a cosmological background, where the scalar fields depend on time,
\begin{equation}
\Upsilon = K_{ij^*} \left( \chi^i \partial_0 \varphi^{j^*} +  \chi^{j^*} \partial_0 \varphi^i \right) \;,
\end{equation}
where the indices run over the number of chiral superfields in the model, $\left( \varphi^i ,\, \chi^i \right)$ are the scalar and fermion components of a chiral complex multiplet (where
$\chi^i$ is a left-handed fermion), while $\left( \varphi^{i^*} ,\, \chi^{i^*} \right)$ are their conjugates.

For simplicity, we consider the case of two chiral superfields, although this can be immediately generalized to an arbitrary number. In this case, the coupled equations for $\theta$ and $\Upsilon$ form a closed system. To write these equations, we introduce the following quantities (that we express through the scalar product defined above)
\begin{eqnarray}
\alpha &\equiv& \rho + 3 m_{3/2}^2 = \dot{\varphi} \cdot \dot{\varphi}^* + F \cdot F^* \;, \nonumber\\
\alpha_1 &\equiv& p - 3 m_{3/2}^2 = \dot{\varphi} \cdot \dot{\varphi}^* - F \cdot F^* \;, \nonumber\\
\alpha_2 &\equiv& 2 \frac{\partial}{\partial t}
\left[ {\rm e}^{K/2} \, W \, P_L + {\rm e}^{K/2} \, W^* \, P_R \right] \nonumber\\
&=& 2 \left( \dot{\varphi} \cdot F^* \, P_L +
\dot{\varphi}^* \cdot F \, P_R \right) \nonumber\\
&& \quad\quad + {\rm e}^{K/2} \left( \dot{\varphi}^i \frac{\partial K}{\partial \varphi^i} -  \dot{\varphi}^{*i} \frac{\partial K}{\partial \varphi^{*i}} \right) \left( W^* \, P_R - W \, P_L \right) \;. \nonumber\\
\end{eqnarray}
In these expressions $\rho$ and $p$ denote, respectively, the energy density and pressure of the background scalars, while $P_{L/R}$ are the left- and right-handed chiral fermion projection operators. These expressions considerably simplify along real solutions for the scalar fields
\begin{eqnarray}
{\rm real \; scalars}: \;&& \alpha =
\dot{\varphi} \cdot \dot{\varphi} + F \cdot F \;, \nonumber\\
&& \alpha_1  = \dot{\varphi} \cdot \dot{\varphi} - F \cdot F \;, \nonumber\\
&& \alpha_2 = 2 \, \dot{\varphi} \cdot F \;.
\label{alphas-real}
\end{eqnarray}
These expressions apply to the models considered in the previous sections, that have real background solutions. In the following we employ the simplified expressions (\ref{alphas-real}).

The closed system of equations for $\theta$ and $\Upsilon$ is
\begin{equation}
\left( \gamma^0 \, \partial_0 + i \gamma^i \, k_i \, N + M \right) X = 0 \;\;\;\;\;,\;\;\;\;\; X = \left( \begin{array}{c}
{\tilde \theta} \\ {\tilde \Upsilon}
\end{array} \right) \;,
\label{eqs-theta-Y}
\end{equation}
where ${\tilde \theta}$ and ${\tilde \Upsilon}$, are canonically-normalized fields, related to the original fields by
\begin{equation}
\theta \equiv \frac{2 i \gamma^i \, k_i}{\left( \alpha \, a^3 \right)^{1/2}} \, {\tilde \theta} \;\;\;,\;\;\;
\Upsilon \equiv \frac{\Delta}{2} \left( \frac{\alpha}{a} \right)^{1/2} \, {\tilde \Upsilon } \;,
\end{equation}
(with $\Delta$ to be defined shortly), and where
\begin{equation}
N = \left( \begin{array}{cc}
- \frac{\alpha_1}{\alpha} - \gamma^0 \, \frac{\alpha_2}{\alpha} & - \gamma^0 \, \Delta \\
- \gamma^0 \, \Delta & - \frac{\alpha_1}{\alpha} + \gamma^0 \, \frac{\alpha_2}{\alpha}
\end{array} \right) \;,
\label{N}
\end{equation}
while the expression for $M$ is not important for the present discussion, and can be found in \cite{Nilles:2001fg}.

Disregarding the presence of the field $\Upsilon$ amounts to the system studied in
\cite{Kolb:2021xfn,Kolb:2021nob}. The square of the gravitino sound speed would then be given by the square of the $N_{11}$ element (we note that, due to the signature we have chosen, $\gamma^0$ is anti-hermitian),
\begin{equation}
N_{11} \, N_{11}^\dagger = \frac{\left\vert \alpha_1 \right\vert^2 + \left\vert \alpha_2 \right\vert^2}{\alpha^2} = c_s^2 \;.
\end{equation}
Namely, using the expressions (\ref{alphas-real}) leads precisely to the sound speed (\ref{vs2}). The complete system however has also off-diagonal elements, with %
\begin{equation}
\Delta = \sqrt{1-c_s^2} \;.
\end{equation}
Therefore, when the coefficient $c_s^2$ vanishes,
\begin{equation}
c_s^2 = 0 \;\; \rightarrow \;\; \Delta = 1 \;\;,\;\; N =  \left( \begin{array}{cc}
0 & - \gamma^0  \\
- \gamma^0  & 0
\end{array} \right) \;,
\end{equation}
leading to a non-singular matrix $N$, and therefore to a nonvanishing sound speed for the physical eigenstates of the system.
Consequently {\em none} of the models discussed in Section 3 have catastrophic production of gravitinos. Problems can only arise if $\Upsilon$ can be ignored as is the case when a second, orthogonal nilpotency condition, is applied as discussed in the next section. Then, the only problematic models would be those defined in Eqs.~(\ref{kolb-Lambda}) and (\ref{staro1}), in the case where the inflaton multiplet $\Phi$ is subject to the additional orthogonal constraint $S ({\Phi} - \overline{\Phi}) = 0$. Indeed, this additional constraint removes the inflatino from the spectrum (hence $\Upsilon = 0$) and the speed of sound in these models hits zero at some point during the inflationary evolution. As we will see, such models seem suspicious from the viewpoint of a fundamental theory of gravity.

\section{Models with Orthogonal Nilpotent Superfields}
\label{sec:nil}

In the cases considered in this paper, the goldstino $G$ belongs to a chiral multiplet and has a scalar superpartner (the sgoldstino), which, once supersymmetry is broken,  acquires a non-supersymmetric mass.
Decoupling the sgoldstino by giving it an {\it infinite} mass leads to  a non-linear realization of supersymmetry.
A particularly simple way non-linear realization can be obtained is by imposing the nilpotent constraint
\cite{Rocek:1978nb,Komargodski:2009rz,Antoniadis:2014oya,Ferrara:2014kva,Kallosh:2014via,DallAgata:2014qsj,Dudas:2015eha,Ferrara:2015gta,Ferrara:2015cwa}
\begin{equation}
\label{X2}
S^2 = 0 \, .
\end{equation}

When supersymmetry is broken by means of a non-trivial $F^S \neq 0$, the constraint is solved by
\begin{eqnarray}
S = \frac{G^2}{2F^S} + \sqrt 2 \theta G + \theta^2 F^S \, .
\label{nil1}
\end{eqnarray}
Here and in what follows, we discuss the constraints at the level of global supersymmetry for simplicity. The generalization to supergravity can be found in the literature \cite{Antoniadis:2014oya}-\cite{DallAgata:2016syy}.
The constraint (\ref{X2}) can be interpreted as the infrared limit of a very heavy sgoldstino. This can be obtained starting from a microscopic Lagrangian of the type \cite{Komargodski:2009rz}:
\begin{equation}
K = |S|^2 - \frac{1}{\Lambda^2} |S|^4 \, , \qquad \ W = W_0 + W_1 S \, ,  \label{nil2}
\end{equation}
in the limit $\Lambda \rightarrow 0$. Indeed, the sgoldstino mass
$m_S^2 = 4 \frac{{F^S}^2}{\Lambda^2}$ is sent to infinity in this limit, leading to a nonlinear realization of supersymmetry in the IR. The limit $\Lambda \rightarrow 0$ has its limitations
\cite{Dudas:2016eej}, since it implies  field-theory dynamics in some heavy sector, which after decoupling, leaves behind the ``strong stabilization" term $\frac{1}{\Lambda^2} |S|^4$. Modulo these subtleties, the UV Lagrangian  (\ref{nil2}) contains only two derivatives and is pretty standard.

The situation is different for the orthogonal constraint on the chiral superfield ${\Phi}$ that removes the imaginary part of the scalar, the fermion, and the auxiliary field \cite{Komargodski:2009rz,Ferrara:2015tyn,Carrasco:2015iij,DallAgata:2015zxp,Kallosh:2016hcm,DallAgata:2016syy}:
\begin{equation}
S ({\Phi} - \overline{\Phi}) = 0 \ . \label{nil3}
\end{equation}
It was shown in \cite{DallAgata:2016syy}  that (\ref{nil3}) is equivalent to the following set of constraints
\begin{eqnarray}\label{nil4}
 |S|^2 ({\Phi} - \overline{\Phi} )&=& 0 \, ,  \\[2mm]
\label{nil5}
 |S|^2 \overline D_{\dot \alpha} \overline{\Phi} &=& 0\, , \\[2mm]
 \label{nil6}
 |S|^2 \overline D^2 \overline{\Phi} &=& 0 \, .
\end{eqnarray}
Each constraint above eliminates one component field:
Eq.~(\ref{nil4}) eliminates the scalar, Eq.~(\ref{nil5}) eliminates the fermion, whereas Eq.~(\ref{nil6}) eliminates the auxiliary field in the $\Phi$ multiplet.
The constraint \eqref{nil3} can be obtained starting from a microscopic Lagrangian containing three additional terms \cite{DallAgata:2016syy}, which generate non-supersymmetric masses for the component fields that we remove:
\begin{eqnarray}
&& 	 \int \! d^4 \theta \Big{[} \frac{m^2_b}{2f^2} |S|^2 ({\Phi} - \overline{\Phi} )^2
	- \frac{g_{F^{\Phi}}}{f^2}  |S|^2 D^2 {\Phi} \overline D^2 \overline {\Phi } \Big{]} \nonumber \\
&& 	- \frac{m_\zeta}{2 f^2} \int \! d^4 \theta \Big{[}  |S|^2  D^{\alpha} {\Phi} D_\alpha {\Phi} + c.c. \Big{]} \ . \label{nil7}
\end{eqnarray}
In (\ref{nil7}), $f$ can be taken to be (by convention) the supersymmetry breaking scale in the vacuum $f= \langle F^S \rangle$, $m_b$ and $m_{\zeta}$ are mass parameters for the decoupling scalar and fermion, respectively, $g_{F^{\Phi}}$
is a dimensionless coupling, $D_\alpha$ denotes a covariant derivative in superspace and $D^2 =D^\alpha D_\alpha$, see e.g. \cite{Wess:1992cp}.
Notice that only the first term in (\ref{nil7}) is a standard correction to the K\"ahler potential, whereas the second and the third terms contain higher-derivatives.
The limit $m_b \rightarrow \infty$, $g_{F^{\Phi}} \rightarrow \infty$ and $m_\zeta \rightarrow \infty$ generate infinitely large masses to the scalar and fermion and an infinite coefficient to the auxiliary field $F^{\Phi}$. In this limit,  the superspace equations of motion for the chiral superfield ${\Phi}$ are dominated by
\begin{eqnarray}
&& 	\overline D^2 \left\{ \frac{m^2_b}{f^2} |S|^2 ({\Phi} - \overline{\Phi} )
	+\frac{m_\zeta}{f^2} D^\alpha ( |S|^2 D_\alpha {\Phi} ) \right.
	\nonumber \\
&& \quad\quad \left.	- \frac{g_{F^{\Phi}}}{f^2} D^2 (|S|^2 \overline D^2 \overline {\Phi}) \right\}  = 0 \ .\label{nil8}
\end{eqnarray}
This indeed reproduces the constraints (\ref{nil4})-(\ref{nil6}) and therefore  (\ref{nil3}): multiplication with $S \overline S$ leads to (\ref{nil6}); the multiplication with $D_\beta S \overline S$ gives (\ref{nil5}); and finally, using (\ref{nil6}) and (\ref{nil5}) in (\ref{nil8}) then multiplying with $\overline S$ leads to (\ref{nil4}).

As emphasized already,  in contrast to the nilpotent UV Lagrangian
(\ref{nil2}), the action (\ref{nil7}) contains higher derivatives. As such, it is qualitatively different and the orthogonal constraint cannot therefore be obtained, to our knowledge, starting from a standard two-derivative action in the UV. In particular, if the chiral superfield $S$ was not nilpotent, the second term in Eq. (\ref{nil7}) proportional to $g_{F^\Phi}$ could  introduce ghosts into the theory. This is a special property of the decoupling procedure, and could signal that such constraints could (but not necessarily) come from a sick UV theory.
This could explain the problems with the slow gravitinos in some inflationary models using the orthogonal constraint.

We conclude this section with some general properties and implications of supersymmetric models with a second degree nilpotency condition, $S^2 = 0$, and the orthogonality constraint, $S(\Phi - \bar{\Phi}) = 0$. It follows from these constraints that if we choose to work in the unitary gauge the goldstino and inflatino fields are absent in these models. If we consider a supergravity model with a K\"ahler potential and superpotential of the form
\begin{equation}
    K = -\frac{\left(\Phi - \bar{\Phi} \right)^2}{2} + S \bar{S} \, ,
\end{equation}
and
\begin{equation}
    \label{sup1}
    W (\Phi, S) \; = \; f(\Phi) S + g(\Phi) \, ,
\end{equation}
so that the K\"ahler potential possesses a shift symmetry, and vanishes when the field constraints are imposed, $K(S, \bar{S}; \Phi, \bar{\Phi})|_{S = \bar{S} = \Phi - \bar{\Phi} = 0} = 0$. The orthogonality constraint ensures that the covariant derivative $D_{\Phi}W$ is absent in the effective scalar potential, which is then given by \cite{Ferrara:2015tyn,Carrasco:2015iij}
\begin{equation}
    \label{pot1}
    V  \; = \; e^{K} \left( K^{S S^{*}} |D_S W|^2 - 3|W|^2  \right) = f^2 - 3 m_{3/2}^2 \, ,
\end{equation}
where the gravitino mass is
\begin{equation}
    \label{gravmass}
    m_{3/2} \; = \; |g(\Phi)| \, .
\end{equation}

Note that the sound speed given in Eq.~(\ref{vs2}) should not be used in this case, since $F^\phi = 0$ for the bosonic part, while $D_\phi W \ne 0$.  Rather, one can use
Eq.~(\ref{Eq:generic32}) and the potential in Eq.~(\ref{pot1}) with Eq.~(\ref{gravmass}) to obtain
\begin{eqnarray}
    c_s^2 & = & \frac{\left(f(\Phi)^2 -\dot{\Phi}^2 \right)^2}{\left(f(\Phi)^2 + \dot{\Phi}^2 \right)^2} + \frac{(2 g'(\Phi) \dot{\Phi})^2}{\left(f(\Phi)^2 +\dot{\Phi}^2 \right)^2} \nonumber \\
    & = & 1- 4 \frac{\dot{\Phi}^2}{(f^2 + \dot{\Phi}^2)^2} \left( f^2 - g'^2 \right)
    \, ,
      \label{soundspeed}
\end{eqnarray}
where $g'(\Phi) = \partial g/ \partial \Phi$. If we want to ensure that the sound speed $c_s > 0$ does not vanish, we must impose the constraint
\begin{equation}
    \label{conspos}
    \left( f(\Phi)^2 - \Dot{\Phi}^2 \right)^2 + \left(2 g'(\Phi) \dot{\Phi} \right)^2 > 0 \, .
\end{equation}
The above quantity is non-negative  at all times in moduli space and this condition is only violated when $f(\Phi) = \pm \dot{\Phi}$ and $g'(\Phi) = 0$. If the gravitino mass is constant at all times, this implies that $g'(\Phi) = 0$ and the constraint~(\ref{conspos}) will be violated when the first term vanishes.

While it is straightforward to construct models for which the gravitino mass is not constant,
additional problems quickly ensue. Consider, for example, a superpotential defined by
\begin{equation}
    \label{funcs1}
    f = f_0 + \sqrt{3}g, \qquad g = \frac{1}{2\sqrt{3}} \left(a f_0  + \tilde{m} \right) \, ,
\end{equation}
where $\tilde{m}$ is associated with the weak scale, and the parameter $a$ is an arbitrary positive number. At the minimum of the potential, the gravitino mass is of order ${\tilde m}$. Any variation in a quantity this small will still lead to a sound speed which is also very small. For $a \ne 0$, the variation of the gravitino mass may be comparable to the potential.  The effective scalar potential~(\ref{pot1}) becomes
\begin{equation}
    \label{pot2}
    V = (a + 1) f_0^2 + \tilde{m} f_0 \, .
\end{equation}
If we choose
\begin{equation}
 f_0  =  \frac{\sqrt{3}}{2 \sqrt{a + 1}} m \left(1 - e^{-\frac{2}{\sqrt{3}}\Phi} \right) \, ,
\end{equation}
and use the canonically-normalized field $\Phi = \phi/\sqrt{2}$, we recover approximately, for $\tilde m \ll m$, the Starobinsky inflationary potential~(\ref{staropot}).
However, during each oscillation, when $\Phi = 0$,
$f^2 - g'^2 = -a^2/12(1+a)$.
As a result the sound speed exceeds 1, signalling a sickness in the theory.
The Cauchy-Schwarz inequality noted above is no longer
guaranteed to hold as the sound speed is now
\begin{equation}
c_s^2 = 1 - \frac{4}{
\left(\vert\dot{\Phi}\vert^2 + \vert F^S \vert^2 \right)^2} \,
\left\{
 |\dot{\Phi}|^2  |F^S|^2  -
\left\vert \dot{\Phi}  e^{K/2} D_{\Phi} W  \right\vert^2 \right\}  \;,
\label{vs3}
\end{equation}
and there is no such inequality between $|\dot{\varphi}|^2  |F|^2 = |\dot{\Phi}|^2 |F^S|^2$ and $\left\vert \dot{\varphi}  \cdot F^*  \right\vert^2 = |e^{K/2} \dot{\Phi} D_\Phi W|^2 $,
 since we can no longer use Eq.~(\ref{DWF}) to relate $F^\Phi$ and $D_\Phi W$.
Thus pathologies might arise whenever $|e^{K/2} D_\Phi W| > |F^S|$.
Such behavior was also seen in \cite{Kolb:2021xfn}.  We are not sure if this behavior is a remnant of the lack of two derivative UV origin stemming from the orthogonal constraint.

\section{Conclusions}
\label{sec:conclusions}

If supersymmetry is realized below the Planck scale, it is natural to construct models of inflation in the context of supersymmetry/supergravity. Particle production and reheating after the period of exponential expansion is an essential component of any model of inflation, and in the context of
supersymmetry, the production of gravitinos must be considered.
If the gravitino is not the lightest supersymmetric particle, it is expected to be unstable. Because of its gravitational coupling, its lifetime can be quite long (for weak scale gravitino masses), causing a host of potential problems \cite{sw}.

In general, the production of gravitinos following inflation is complicated by the possibility that the field content of the longitudinal mode comprised of the goldstino may be different during inflation from that in the vacuum. Indeed,
supersymmetry might be broken during inflation,
and the longitudinal component of the gravitino may be identified with the inflatino. Thus, models
with substantial non-perturbative particle production \cite{Kallosh:1999jj,Giudice:1999yt,Giudice:1999am,Kallosh:2000ve}, may be producing an inflatino rather than a gravitino \cite{Nilles:2001ry,Nilles:2001fg}.
The inflatino problem is generally model dependent
and is non-existent in certain constructions \cite{Nilles:2001my}.

An additional problem associated with gravitinos has been discussed recently \cite{hase,Kolb:2021xfn,Kolb:2021nob} in connection with a vanishing sound speed for gravitinos.
It was argued that when $c_s \rightarrow 0$, the non-perturbative production
of high-momentum modes in the helicity 1/2 mode is unsuppressed.  Indeed, there are a variety of well-studied models where the gravitino sound speed does vanish. We have discussed several examples for which $c_s \rightarrow 0$ and several for which it does not. That is, while a vanishing sound speed may occur in supergravity models of inflation, it is by no means a necessary consequence. Furthermore, we have shown that even in the models where $c_s \rightarrow 0$, runaway gravitino production is blocked by mixing with the inflatino in a time-dependent background. This is the case in models with and without a nilpotent field associated with supersymmetry breaking.

However, the problem of catastrophic gravitino production may occur in models with a second orthogonal constraint for which the inflatino is removed from the particle spectrum. It is not clear how these models can be extracted from a UV theory involving no more than two derivatives. This argument along with the problem of a vanishing sound speed (or superluminal sound speed when $c_s \ne 0$) may be a constraint on models with constrained fields rather than supergravity models of inflation in general.

\vskip.1in
{\bf Acknowledgments:}
\noindent
The authors want to thank T. Gherghetta, H.P. Nilles, and L. Sorbo for helpful discussions. This project has received support from the European Union's Horizon 2020 research and innovation programme under the Marie Sk$\lslash$odowska-Curie grant agreement No 860881-HIDDeN, by the ANR grant Black-dS-String ANR-16-CE31-0004-01, the CNRS PICS MicroDark, and the IN2P3 Master Project UCMN.
The work of K.A.O.~was supported in part by DOE grant DE-SC0011842  at the University of
Minnesota. The work of MG was supported by the Spanish Agencia Estatal de Investigaci\'on through Grants No.~FPA2015-65929-P (MINECO/FEDER, UE) and No.~PGC2018095161-B-I00, IFT Centro de Excelencia Severo Ochoa SEV-2016-0597, and Red Consolider MultiDark FPA2017-90566-REDC.