# Characterizing the Dark Count Rate of a Large-Format MKID Array

Noah Swimmer1  W. Hawkins Clay1  Nicholas Zobrist1  Benjamin A.
Mazin1,$\dagger$ 1Department of Physics, University of California, Santa
Barbara, California, 93107, USA  *<EMAIL_ADDRESS>
###### Abstract

We present an empirical measurement of the dark count rate seen in a large-
format MKID array identical to those currently in use at observatories such as
Subaru on Maunakea. This work provides compelling evidence for their utility
in future experiments that require low-count rate, quiet environments such as
dark matter direct detection. Across the bandpass from 0.946-1.534 eV
(1310-808 nm) an average count rate of $(1.847\pm 0.003)\times 10^{-3}$
photons/pixel/s is measured. Breaking this bandpass into 5 equal-energy bins
based on the resolving power of the detectors we find the average dark count
rate seen in an MKID is $(6.26\pm 0.04)\times 10^{-4}$ photons/pixel/s from
0.946-1.063 eV and $(2.73\pm 0.02)\times 10^{-4}$ photons/pixel/s at
1.416-1.534eV. Using lower-noise readout electronics to read out a single MKID
pixel we demonstrate that the events measured while the detector is not
illuminated largely appear to be a combination of real photons, possible
fluorescence caused by cosmic rays, and phonon events in the array substrate.
We also find that using lower-noise readout electronics on a single MKID pixel
we measure a dark count rate of $(9.3\pm 0.9)\times 10^{-4}$ photons/pixel/s
over the same bandpass (0.946-1.534 eV) With the single-pixel readout we also
characterize the events when the detectors are not illuminated and show that
these responses in the MKID are distinct from photons from known light sources
such as a laser, likely coming from cosmic ray excitations.

††journal: oe††articletype: Research Article

$\dagger$ http://mazinlab.org

##  1 Introduction

Microwave Kinetic Inductance Detectors (MKIDs) are superconducting
microresonators with the ability to measure individual photons[1]. These light
sensitive detectors can measure photon arrival times with microsecond
precision and their unique detection mechanism also allows them to measure the
energy of each incident photon. Since they utilize a different readout scheme
than conventional image sensors such as charge-coupled devices (CCDs), MKIDs
have the additional benefit of not being susceptible to read noise (stemming
from the conversion of electrons from a CCD potential well into a voltage
signal before being quantized and processed) or dark current (when thermal
electrons accumulate in the potential well of the CCD and are counted as part
of the signal upon readout despite their different origin)[2, 3, 4, 5, 6].
This, along with their photon counting ability makes MKIDs exceptional
detectors for astronomy in photon starved regimes [7, 8, 9].

Current-generation MKIDs have been designed to be sensitive to photons in
different ranges such as for ultraviolet, optical, and near-infrared (UVOIR)
astronomy and X-ray detection. They also offer straightforward ways to tune
their sensitivity to higher- or lower-energy bandpasses. This ability opens
opportunities for MKIDs to be used as detectors for new physics applications
such as the search for dark matter. In ‘photon-starved’ regimes, where sources
emit very few photons, it is of high importance to adequately characterize the
performance of the detectors to ensure that each photon detected is ‘real’,
i.e. light from the astronomical source being observed rather than from errant
sources such as thermal blackbody radiation from inside a cryostat.

In this work we aim to characterize “dark counts” measured by a large-format
MKID array. These are events that are registered as photons by the detector
when it is not exposed to a light source. These dark counts differ from those
of conventional semiconductor detectors because of their origin. Semiconductor
detectors register false counts due to dark current and read noise. Dark
current is the generation of thermal electrons in the material that are
captured by the detector’s potential well and counted as part of the signal
while read noise is the noise that is added to the measured signal from charge
photon-to-voltage conversion and signal processing such as analog to digital
conversion. In MKIDs false triggers may stem from noise in the room-
temperature readout[4], blackbody photons from the environment, or more
complex sources.

##  2 MKID photon measurement

Figure 1: A single photon event measured by an MKID pixel with readout
sampling at 0.8 MHz. The x-axis shows time in microseconds and y-axis shows
the detector phase response measured in radians.

Figure 2: 2 classes of ‘bad’ photon events that can be triggered in the MKID
readout. (Left) A phase time stream (phase measured in radians) taken when
noise in the data caused the readout to trigger as if a photon hit the
detector although clearly none did. (Right) A second photon arriving on the
tail of a previous one. This contaminates the analysis of the first and,
depending on the dead time, the readout may not trigger at all on the second.

Each MKID pixel – “pixel” and “detector” may be used interchangeably – is a
superconducting LC-circuit which is excited at its resonant frequency by a
probe tone using room temperature readout electronics. When an incident photon
is absorbed by an MKID pixel, the energy from the photon breaks Cooper pairs
in the resonator which causes the inductance of the resonator to increase.
This increase in inductance is measured as a change in the phase of the
microwave probe tone. The probe tone for each MKID pixel is typically sampled
at 1 MHz, giving microsecond timing resolution [10, 2, 4].

For an MKID pixel to detect a photon, the photon event must cause the phase of
the detector to increase beyond a minimum threshold. For a given pixel this
threshold is calculated by measuring the phase (in radians) while not
illuminated. This allows a phase noise to be measured, which we find does not
typically exceed $\sigma_{\phi}\sim$0.15 radians in good-performing pixels.
The threshold for the given resonator is then set to be 6$\sigma_{\phi}$ away
from the average phase value – typically 0 radians. In other terms, the
threshold for each pixel is set to be

$\phi_{\rm Threshold}=\hat{\phi}+6\sigma_{\phi}\approx 6\sigma_{\phi}$ (1)

where the right hand approximation holds when $\hat{\phi}\approx$0\. This is
the method by which each pixel’s photon detection threshold is calculated for
both electronic readout systems described in this paper (Sections 4 and 5).
Since the phase noise is Gaussian and a 6$\sigma_{\phi}$ threshold is used,
that means that there is less than a 1 in 500,000,000 chance that the phase
will fluctuate above the threshold at any point when the phase is measured.
The process for determining the threshold for a pixel is described in detail
in [11]. Additionally, each pixel’s phase response to photons of different
energies within the bandpass of interest is measured when the array is
calibrated (Section 3.3). For the MKID resonators used in this experiment we
find that the typical phase response for the lowest-energy photons in the
bandpass (0.946 eV, $\lambda$=1310 nm) is $\phi_{\rm 0.946eV}=-1.4\pm 0.2$
radians and the typical phase response for the highest-energy photons (1.534
eV, $\lambda$=808 nm) is $\phi_{\rm 1.534eV}=-2.2\pm 0.3$ radians. This
demonstrates that the phase noise is unlikely to ever swing sufficiently high
to cause a resonator to trigger on an event within the calibrated bandpass.

It is also instructive to calculate the signal-to-noise ratio (SNR) for photon
pulses seen by each resonator. The resolving power $\mathcal{R}=E/\Delta E\sim
E/2.355\sigma_{\phi}$, where $E$ is the energy of a photon event and is
proportional to the phase pulse height, meaning that the SNR $\sim
E/\sigma_{\phi}$ (see Section 3.3). Rearranging, one finds that the SNR $\sim
2.355\mathcal{R}$. This means that for a resonator with an $\mathcal{R}=4.5$ –
a typical value in this experiment – the SNR for a photon pulse will be about
10.6, again demonstrating that a resonator’s phase response to a photon will
be significantly higher than the phase noise itself.

An example of a single photon event is shown in Figure 1. When a photon is
measured the readout records the time at which the photon struck and the
height of the peak of the pulse. The height of the pulse (which is
proportional to the change in the resonator’s phase) is related to the energy
of the incident photon. Therefore, a photon measured using the MKID readout
intrinsically gathers information about both the time a photon arrived and its
energy. By performing an energy calibration with lasers of known wavelength,
these measured pulse heights can be converted to energies of the incident
photons within the calibrated range.

Figure 2 shows several types of events measured by a single detector that can
contaminate a dataset. The first occurs when noise in the system causes the
readout to register a photon when there is clearly none present. This may
occur when there is a small random spike in the phase data or a very low
energy photon hits the detector. These “noise” triggered events are removed in
post-processing as their peak pulse heights fall outside of the calibrated
energy bandpass and can therefore be removed when cutting photons that are
outside this energy range. This cut will exclude any of the “noise” triggers
that initially registered as a photon. The second is when multiple photons are
caught riding on the tail of the initial photon registered by the readout.
This can be mitigated by decreasing the detector ‘dead time’, the time after a
photon is measured before the readout can read a second photon. In the digital
readout (Section 4) the first photon is counted while the second (and beyond)
photon is ignored completely, while in the analog readout (Section 5) one can
manually calibrate these and split up timestreams that have multiple photon
events within them, including both if desired or treating them similarly to
the digital readout and not registering the rest.

##  3 Experiment overview

###  3.1 Optics

Figure 3: Schematic of the optical path from the laser at room temperature to
the MKID array and of the idealized analog readout with parametric
amplification. Laser light is carried down a multimode fiber that is inserted
into the input port of a collimator. The collimated light is incident on a
microlens array which then focuses the light onto each MKID pixel. The signal
comes from synthesizer A, and the pump tone for the TWPA comes from
synthesizer B. Each of the resistors to ground represents a 50$\Omega$
termination. (Inset) A single MKID pixel showing the ideal focus spot of light
onto the photosensitive inductor portion of the detector.

The optical path to put light onto the MKID array is relatively simple. One of
five lasers within the sensitivity range of the devices (808 nm, 920 nm, 980
nm, 1120 nm, 1310 nm) can be inserted into an integrating sphere which has an
output to a multimode fiber at room temperature. The fiber is then inserted
into a port in the dilution refrigerator where it is routed to directly to a
collimator at the same temperature as the MKID device. The fiber has
transmission greater than 90$\%$ at all of the laser wavelengths above. within
the The collimator is oriented so the collimated light shines directly onto a
microlens array (MLA) which focuses the light onto the photosensitive inductor
of each MKID pixel. The collimator creates a spot greater in size than the
MKID array, meaning that the distribution of light from the lasers is
spatially uniform across the MKID pixels. A schematic of the optical path
within the fridge is shown in Figure 3, while the inset shows a spot of light
focused on a single MKID pixel.

To prevent potential stray photons from the inside of the dilution
refrigerator from hitting the MKID array, a special lid was created so that
the fiber collimator mounts directly to the box which houses the MKID array,
preventing any unwanted photons from entering the window and hitting a
detector.

###  3.2 MKID array

Figure 4: A 10-feedline MEC-style MKID array with 20,440 pixels. The lid with
microlens array has been removed so that the MKID array itself can be seen.
The MKID digital readout (Section 4) combined with the dilution refrigerator
setup (Figure 3) allow for up to 1 of the microwave feedlines (2044 MKID
pixels) to be read out at a time. In an MKID instrument all 10 can be read out
simultaneously.

The MKID array used in this experiment has 10 microwave feedlines, each with
2044 MKID resonators coupled to it for a total of 20,440 pixels. This device
has the same design as the MKID arrays currently in use with the MKID
Exoplanet Camera (MEC, [12]) which is currently commissioned at the Subaru
Telescope on Maunakea. An image of a MEC-style array can be seen in Figure 4.

In the MEC instrument each of the 20,440 pixels can be read out simultaneously
using the 2nd Generation MKID Digital Readout (see Section 4 for further
details)[4] which provides exquisite spatial resolution while time-tagging
incident photons with microsecond accuracy and is theoretically capable of
resolving each photon’s energy to within several percent[10] although the
resolving power may be degraded by noise from hard-to-control experimental
sources such including electrical noise in the room-temperature readout
electronics, magnetic fields and Johnson noise in the dilution refrigerator,
two-level system noise in the detectors, and phase measurement noise. The
resolving power may also be affected by strong radio frequency interference
(RFI) from sources such as 5 GHz Wi-Fi that are close in frequency space to
the resonant frequencies of the MKIDs (4-8 GHz).

While mounted in an instrument, and MKID array has a lid to protect stray
light that may be incident on it and a microlens array (MLA) which is designed
to focus light onto the photosensitive region of each MKID resonator. Figure 4
does not show the lid+MLA so that the array itself may be seen for clarity.

###  3.3 Array calibration

Energy (eV) | Wavelength (nm) | Median $\mathcal{R}_{1}$ | Median $\mathcal{R}_{2}$
---|---|---|---
0.946 | 1310 | 4.5 | 4.9
1.107 | 1120 | 4.4 | 4.5
1.265 | 980 | 4.4 | 4.4
1.348 | 920 | 4.4 | 4.5
1.534 | 808 | 4.6 | 4.5
Table 1: The median resolving power of all resonators measured at each
calibration laser energy (wavelength). Columns 1 and 2 show the energy and
wavelength of each laser used for calibration. Column 3 shows the median
resolution values of all resonators taken prior to data collection while
column 4 shows the same values taken 3 days after, following the conclusion of
data collection.

The resolving power of an MKID pixel for a given photon energy is given by
$\mathcal{R}=E/\Delta E$. Here, $E$ is the energy of the incident photons –
typically from a monochromatic laser – and $\Delta E$ is measured by
determining the full width at half maximum (FWHM) of the distribution of
photon energies recorded by the pixel. To measure the resolving power of each
resonator, five lasers are shined across the array, one at a time. The
responses of each resonator are recorded and used to find its resolving power
at each of the different laser wavelengths[11].

819 MKID pixels were initially identified before any data collection. An
energy calibration dataset was taken prior to and after data acquisition to
assess the stability of each pixel’s response to photons of a given energy.

We require that each resonator was successfully energy calibrated in both
datasets. This means that each pixel was marked by the energy calibration
software as being successfully calibrated at each of the 5 laser energies.
This cut removed 40 pixels, leaving 779 (95.1$\%$) of the original 819. This
cut was made for two reasons. The first being that if the MKID Data Reduction
Pipeline [11] is not able to identify a resonator in one of the two
calibration datasets, we cannot confirm that it stayed stable through the
duration of the data collection. This was responsible for removing 14 of the
40 pixels cut. The second is that even if a resonator was identified in both
calibration datasets, it must be able to be successfully energy calibrated at
all 5 laser energies. We have seen that resonators that only pass 3 or 4 of
the laser energies are worse performing and typically see many low-energy
‘noise triggered events’ that make such pixels unreliable. This led to the
other 26 of the 40 removed pixels being cut. The median $\mathcal{R}$ values
at each energy are shown for each calibration dataset in Table 1.

These 779 best performing pixels that remained after these cuts are those
whose data will be used for the dark count analysis in Section 4.2. In Section
5 a single one of these resonators will be used to characterize the nature of
the photons that are seen.

###  3.4 Electronic readouts

This investigation used 2 separate readout systems. The first readout system
is the MKID Digital Readout described in [4] which used to read a substantial
number of MKID pixels on one (or more) microwave feedline(s). The second
system uses an analog readout scheme similar to that described in [3, 6] to
read out a single MKID pixel. This system ensures lower readout noise and the
ability to record photon phase timestreams . We will discuss the former in
Section 4 and the latter in Section 5.

##  4 MKID digital readout: many-detector measurement

###  4.1 Experimental setup

The full-array readout in this experiment used the second generation MKID
digital readout[4] which is divided between 3 temperature stages where most of
the large electronics are at room temperature. The remainder were housed in a
BlueFors Dilution Refrigerator which cools them to 4K or 100 mK, depending on
the component. The internal schematic for the fridge is seen in Figure 3.

The room-temperature components are discussed in detail in [4] but are briefly
described here for context and clarity. A 2nd-generation MKID readout board
has a Xilinx Virtex-7 FPGA (Field Programmable Gate Array) controls Analog-to-
Digital Converter/Digital-to-Analog Converter (ADC/DAC) boards which will
generate (DAC) and take in (ADC) the in-phase (I) and quadrature (Q)
components of each probe tone. It also contains ROACH-2 (Reconfigurable Open
Architecture Computing Hardware) boards developed by CASPER (Collaboration for
Astronomy Signal Processing and Electronics Research), which includes all of
the core digital signal processing for the readout system[13, 14]. This
includes the channelization of probe tones, filtering, and photon pulse
detection. An RF/IF board is also used for conversion of probe tones from the
IF band (-1 to 1 GHz) up to the RF band (4-6 or 6-8 GHz) so that up to 1022
MKID resonators can be read out at their resonant frequencies. Each readout
cartridge contains two sets of readout boards - one to read out pixels with
resonant frequencies from 4-6 GHz and the other to read out pixels with
resonant frequencies from 6-8 GHz - to provide the capability to read out all
of the 2044 MKID pixels on a single microwave feedline. For this experiment we
chose to only read out the higher-frequency half of an MKID feedline from 6-8
GHz to reduce potential noise from boards running on the same cartridge and to
best match the bandpass of the parametric amplifier.

Two 4-meter RF coaxial cables are then attached between the readout cartridge
and the dilution fridge to send the signal to the MKID array and carry the
output back to the readout cartridge to be read out.

Internally, the signal is sent from room temperature to 100 mK using cryogenic
RF coax cables that are heat sunk at intermediate temperature stages to reduce
heat flow to the MKIDs. At the 4K stage there is a 20 dB attenuator to
attenuate Johnson noise from the room temperature input. A second 20 dB
attenuator is added at the 100 mK stage to further reduce Johnson noise from
the 4K stage and to account for the amplification of the signals on the output
side of the array. The probe tones are then sent through the MKID device,
exciting the resonators of a single MKID array microwave feedline. On the
output side the signal is sent through a traveling wave parametric amplifier
(TWPA111Further work is being conducted to characterize the performance of a
TWPA while reading out many MKID pixels and will be the subject of a future
publication.). This is a wideband (its maximal gain occurs between 6-8 GHz),
high-power amplifier capable of reading out photon events with near quantum-
limited amplifier noise [15, 3]. The signal is further amplified by a Low
Noise Factory (LNF) High Electron Mobility Transistor (HEMT) amplifier at 4K.
HEMT amplifiers are often used in cryogenic systems that require relatively
high saturation power, significant dynamic range requirements, and a wide
bandpass (for MKIDs, 4-8 GHz). The microwave signal is then sent back up to
room temperature and routed back to the readout cartridge so the MKIDs may be
measured.

Using the readout to monitor half of one MKID feedline enables up to 1022
pixels to be read out during this experiment. The number of functioning pixels
compared to the total number of possible pixels is called the pixel yield. The
current-generation 20,440 pixel arrays such as the one used in this experiment
have yields of about 80$\%$. Each pixel in an MKID array is unique and can be
rendered non-functioning for different reasons such as a piece of dust or
residue landing on a pixel and shorting it to ground. Variation in the
thickness of the resonators or their chemical composition can also cause them
to move to unpredictable frequencies, colliding with another resonator in
frequency space and making one or both unusable[12]. The pixel yield of about
80$\%$ leads to 819 of the possible 1022 resonators being read out. These
pixels will form the basis of our analysis of the dark count rate measured by
the MKIDs. We note that this is the first simultaneous readout of a large MKID
array with a parametric amplifier that we are aware of.

###  4.2 Data collection and reduction

Using the MKID second generation digital readout[4] each resonator from half
of an MKID array feedline collected data with no light incident on the
detectors for 86,750 seconds (24.1 hours in total) from 10-13 December 2020
while the dilution refrigerator was regulating the device temperature at 100
mK.

Notably, the data taken using the digital readout is useful because it allows
hundreds to thousands of resonators to be read out simultaneously. This
enables useful characterization of the data that can only be inferred via bulk
properties of the array such as identification and rejection of cosmic ray
hits (which contaminate the data) or ‘flashes’ across the device (whose
origins are currently unexplored but whose effects are nevertheless
contaminating and must be removed).

Figure 5: A timestream showing the number of counts across the 819 resonators
read out using the MKID Digital Readout over a 300-second span. The timestream
is created by binning the photon arrival times using 10 $\mu$s bins (i.e. all
photons measured by the 819 resonators are first sorted by time then grouped
into bins of 10 $\mu$s, meaning the y-axis shows the number of photons that
hit the measured resonators within that short time range). The quiescent count
rate is approximately 0 counts/second, but cosmic rays hitting the array and
other potential system noise cause ‘flashes’ where many pixels register a
photon. The inset shows a single cosmic ray event (marked by the red arrow)
for 200 $\mu$s before and 600 $\mu$s after the peak.

The 86,750 seconds of data taken using the MKID Digital Readout were processed
and reduced using the MKID Science Data Pipeline[11]. The data were split into
smaller chunks of time in order to create photon lists (the ‘base’ format for
MKID instrument data, see [7] for further details) of manageable size.

During this reduction, a cosmic ray calibration is performed by making a time-
based coincidence veto based on significantly more photons being detected
across the array in a short burst above the quiescent count rate. This
mitigates cosmic ray events, which occurs when a high energy particle is
absorbed by the MKID array. When an energetic particle absorbed in the
substrate it may down-convert into a cloud of phonons that spreads out across
the array, depositing energy into resonators as the phonons move away from the
point of absorption. As energy is deposited into each resonator, that
resonator will register a single spurious event. Since the energy from the
energetic particle spreads so quickly over many resonators, this will cause a
spike in the number of counts across the array for a short time before
returning to the normal count rate of the observation in question. A full-
array timestream with many cosmic events can be seen in Figure 5. To prevent
any possible contamination from cosmic events, 5000 $\mu$s and 10000 $\mu$s
are removed from before and after each event, respectively (the asymmetrical
nature is due to there being a ‘tailing off’ behavior while the energy spreads
through the array). This veto leads to a short time range surrounding each
cosmic ray event being removed, ultimately removing $\sim 0.3\%$ of the total
time each resonator was taking data.

For the last quality cut we remove any “hot” pixels remaining. Qualitatively,
a pixel is considered “hot” if – after cosmic rays are removed – it counts
significantly more photons than other pixels. These pixels that register
photons more frequently than the average pixel may become "hot" for reasons
such as having a very low phase threshold, being under- or overpowered,
electrical noise in the readout electronics causing the probe tone for that
individual resonator to become noisy, the resonator being adversely affected
by local electrical or magnetic fields and shifting slightly in frequency,
causing the resonator to go out of calibration, or other more complicated
phenomena leading to a specific resonator triggering more frequently than its
neighbors. The hot pixels cause the distribution of the total number of
photons measured per resonator to be highly positively skewed. To properly
characterize the shape of this distribution and catch hot pixels, a metric
that is robust to outliers must be chosen. To this end, the median of the
number of counts ($\hat{c}$) from each resonator was taken to be the
‘expected’ value of the distribution while the spread is measured using the
astropy mad_std function
(https://docs.astropy.org/en/stable/api/astropy.stats.mad_std.html) to
calculate a robust standard deviation ($\sigma_{c,MAD}$) using the Median
Absolute Deviation (MAD). As shown in equation 2 a pixel is considered hot if
the number of counts $c$ it sees is greater than the median number of counts
measured by all pixels plus 15 times the MAD standard deviation.

$c\geq\hat{c}+15\sigma_{c,MAD}$ (2)

This cut leaves 590 of the 779 pixels that remained from Section 3.3.
Ultimately this represents a cut of 24.3$\%$ of the pixels that were energy
calibrated at all 5 laser energies (or 26.7$\%$ of the initial 819 pixels).

The median resolving power of the remaining 590 MKID pixels across all
wavelengths is $\mathcal{R}\sim 4.54$. For each pixel the photons which fell
between the calibrated energy values and were not removed by the cosmic ray
calibration were divided into equal-width energy bins and the count rate of
photons per second was then calculated at each energy. This allowed the count
rate across pixels to be measured, whose values can be seen in Figure 6 and
Table 2.

###  4.3 Analysis

Figure 6: The average count rate values in photon counts per pixel per second
in each energy bin with 1$\sigma$ error bars measured from the ensemble of
count rates from all remaining pixels.

In Section 4.2 the data collection and reduction was discussed in detail.
After the final subset of pixels was determined the dark count rate could be
calculated. To do this, the count rate in each pixel was measured and the
error on each value calculated using Poisson statistics.

The energy bins were determined by the calibrated bandpass (0.946-1.534 eV, or
1310-808 nm) and the median resolving power $\mathcal{R}\sim 5$. This resulted
in 5 energy bins of 0.118 eV centered at 1.005, 1.123, 1.240, 1.358, and 1.476
eV.

The measured count rate for the unilluminated MKID array ranges from $(2.73\pm
0.02)\times 10^{-4}$ photons/pixel/s at 1.476 eV to $(6.26\pm 0.04)\times
10^{-4}$ photons/pixel/s at 1.005 eV after cosmic ray rejection and other
cleaning steps, respectively. This corresponds to an MKID pixel seeing a low
energy photon roughly every 1600$\pm$10 seconds and a high energy photon every
3660$\pm$30 seconds. Alternatively, over the full bandpass the count rate is
measured at $(1.847\pm 0.006)\times 10^{-3}$ photons/pixel/s. In flux units,
this is $(3.14\pm 0.01)\times 10^{-3}$ photons/pixel/s/eV in energy space or
$(3.68\pm 0.01)\times 10^{-6}$ photons/pixel/s/nm in wavelength space.

Although the dark count rate measured by an MKID pixel is not directly
analogous to the dark current measured by CCDs and EMCCDs (Electron-
Multiplying Charge Coupled Devices) due to the differing origin of the events
it is useful to compare the event rates seen by each. Above, we reported that
the number of dark count events measured per MKID pixel is $(1.847\pm
0.006)\times 10^{-3}$ photons/pixel/s. State-of-the-art CCDs have measured
dark count rates of $1.66\times 10^{-3}$ electrons/pixel/s[16] while EMCCDs
have measured dark current rates as low as $1\times 10^{-3}$
electrons/pixel/s[17]. Assuming that the CCD and EMCCD gain (the ratio of
photons needed to generate 1 electron in the CCD/EMCCD) is 1 – i.e. 1 electron
in the detector corresponds to 1 photon event – then the dark current
generation in these state-of-the-art CCDs and EMCCDs are comparable to the
dark count rates measured by an MKID pixel.

####  4.3.1 Comparison to raw data

In Section 4.2 the data reduction algorithm was discussed. This includes the
removal of photons from cosmic ray events, cleaning of particularly noisy
sections of time, and excluding ‘hot’ pixels that live in a state where they
count significantly more photons than their physical neighbors. To show the
improvement these cuts bring we also calculate the dark count rate with none
of the removal steps performed. The results are shown in Table 2.

Prior to performing any data cleaning, each resonator saw an average of
1493$\pm$2 counts across the calibrated bandpass. Afterwards this number was
reduced to 159.5$\pm$0.5, an improvement of by nearly a factor of 10. While we
were able to reduce the number of counts by almost a factor of 10 for each
energy band, the time cut was not particularly aggressive and only about
0.3$\%$ of the duration from each pixel was removed via the cosmic ray cuts.
This shows that the majority of the counts seen come from spurious events that
can be calibrated out without significant time needing to be removed from the
dataset in question.

| With Reduction Steps | No Reduction Steps
---|---|---
Bin Center | Count Rate ($\times 10^{-4}$) | Total Counts | Count Rate ($\times 10^{-3}$) | Total Counts
(eV) | (photons/pixel/s) | (photons) | (photons/pixel/s) | (photons)
1.005 | 6.26$\pm$0.04 | 54.1$\pm$0.3 | 5.0$\pm$0.1 | 436.7$\pm$0.9
1.123 | 3.68$\pm$0.03 | 31.8$\pm$0.2 | 3.83$\pm$0.09 | 332.2$\pm$0.8
1.240 | 3.06$\pm$0.02 | 26.5$\pm$0.2 | 3.17$\pm$0.08 | 275.2$\pm$0.7
1.358 | 2.75$\pm$0.02 | 23.7$\pm$0.2 | 2.73$\pm$0.07 | 237.1$\pm$0.6
1.476 | 2.73$\pm$0.02 | 23.6$\pm$0.2 | 2.44$\pm$0.07 | 211.6$\pm$0.6
Table 2: The average count rate per pixel and total number of counts per pixel
for the data with the different calibration and cleaning steps (see also
Figure 6) compared to the same quantities without. Without reduction steps,
each pixel took data for 86,750 seconds. After cleaning, each pixel was left
with 86,491 seconds of data.

####  4.3.2 Potential photon sources

The steps taken during the data reduction in Section 4.2 aimed to mitigate
effects from noisy pixels and well-understood noise sources such as cosmic ray
events. However there is still the possibility that there are more complex or
uncalibrated sources that are not well characterized in this system. This can
include things such as electrical noise from cryogenic amplifiers or room
temperature readout electronics, secondary photons stemming from cosmic rays
exciting electrons and causing fluorescence in the fiber optic cable, and
simple blackbody radiation.

Work has been done to characterize the noise characteristics of the MKID
Digital Readout[4] although it is not well understood how electrical noise in
the system translates to spurious triggers on non-photon events. However,
well-behaved MKID pixels are partially characterized by showing low noise,
meaning they will be less susceptible to pixel-specific noise causing a false
photon trigger.

In the event that a cosmic ray is absorbed at a point in the fiber optic cable
or the rest of the optical path it may deposit its energy in that material,
exciting electrons which will then release secondary photons from this
particle being absorbed. In this case, individual photons may be generated in
the optical path which can then be transmitted to the MKID detectors. This
would be a ‘true’ photon detection from an unintended physical source.

The previous two sources of photon detections are both issues that may
contaminate sensitive data in a photon-starved environment but are currently
challenging if not impossible to mitigate. For the first, reducing electrical
noise by preventing ground loops, using low-noise power sources, and working
in an isolated environment will reduce false triggers from electronic noise
but in practice this is nearly impossible to eradicate. We note however that
large scale electrical noise typically affects all resonators simultaneously
and can therefore be removed (using a similar time-coincidence veto as a
cosmic ray) or causes single resonators to become ‘hot’ or ‘cold’ which may
also be handled gracefully in the data reduction pipeline [11]. For the
second, a secondary photon from a cosmic ray may be removed in the data
reduction if its energy is sufficiently far outside the calibrated bandpass of
the detector, but if its energy is within the bandpass then it will be
impossible to remove as it is a single photon event and therefore not subject
to the same time-coincidence veto from when a cosmic ray strikes the detector
directly.

Figure 7: MKID spectrum (blue) compared to a 300 K blackbody spectrum (black)
scaled to the central value of the MKID spectrum shown on a logarithmic scale.
In this bandpass the blackbody spectrum varies over 7 orders of magnitude
while the MKID spectrum remains relatively flat. The central point of the 300
K spectrum is normalized to the central point of the MKID spectrum. Error bars
on the MKID spectrum are sufficiently small that they are contained within the
points themselves.

We will explore the possibility that the photons that the MKIDs are seeing
while purportedly unilluminated are coming from a blackbody radiating within
the dilution refrigerator. Although all precautions have been taken to prevent
stray photons from hitting the detector, photons are incredibly difficult to
insulate against so we examine the possibility that the photons measured in
the dark environment come from a thermal source.

First, we generate blackbody spectra for each of the 4 potential temperature
stages which may be generating blackbody photons that could possibly hit the
MKIDs. These are the 100 mK stage where the array itself and fiber collimator
are mounted and the array is directly exposed to, the walls of the 4 K and 50
K intermediate stages that are used to step down from room to operating
temperature and that are nested around the 100 mK stage, and 300 K, which
represents the blackbody radiation from the ambient environment or the inner
face of the outermost temperature stage of the dilution refrigerator.

Using Planck’s Law

$B_{\lambda}(T)=\frac{2hc^{2}}{\lambda^{5}}\frac{1}{exp\big{(}\frac{hc}{\lambda
k_{B}T}\big{)}-1}$ (3)

we find that the flux density of 100 mK, 4 K, and 50 K blackbody radiation
between $\sim$0.9 eV and $\sim$1.6 eV is sufficiently small that a blackbody
at any of these temperatures would not be expected to produce photons from
0.9-1.6 eV over the duration of the experiment (86,750 seconds). Therefore,
the blackbody spectra representing these temperatures are not included in the
analysis.

The 300 K blackbody spectrum can be seen plotted against the spectrum measured
by the MKID pixels in Figure 7. The spectra are shown in units of log10(Flux
Density), where the flux density was measured/calculated in
ergs/s/cm${}^{2}/\mathrm{\mathring{A}}$. The plot is shown in log scale
because while the MKID spectrum remains relatively flat across the bandpass,
ranging from $(5.22\pm 0.04)\times 10^{-14}$
ergs/s/cm${}^{2}/\mathrm{\mathring{A}}$ to $(3.07\pm 0.02)\times 10^{-14}$
ergs/s/cm${}^{2}/\mathrm{\mathring{A}}$, the 300 K blackbody spectrum varies
over 7 orders of magnitude.

The massive discrepancy in shape of the two spectra show that the photon hits
that are still being measured by the MKID pixels are not generated by a 300 K
blackbody. With this and the fact that the 100 mK, 4 K, and 50 K stages will
not generate any blackbody photons over the calibrated bandpass it is possible
to say that the source of the remaining photons measured using the MKID
digital readout do not come solely from blackbody sources in the environment.

##  5 MKID analog readout: single detector measurements

###  5.1 Experimental setup

The analog readout utilizes the same internal electronics of the fridge, but
externally the 6 foot SMA cables attach to a homodyne readout system
consisting of two Anritsu MG37022A Signal generators, a Weinschel Attenuator
box 8310 Series, an National Instruments-ADC/DAC, and an IQ mixer box. The
function of these devices is the same as in the digital readout case. The
schematic for the analog readout system, and the parametric amplifier, is
shown in Figure 3. Unlike the digital readout, the analog readout supplies
individual frequencies from an Anritsu Synthesizer to probe single resonators
on the array. The analog readout has less noise associated with it compared to
the digital readout which has to make compromises so that it is able to issue
many probe tones while also dealing with limited dynamic range in the ADC
attenuators and precision in its firmware computations.

The primary reason for taking a set of data in the dark with the analog
readout is to analyze the nature of the photon pulses. Despite the fact that
MKIDs are not susceptible to read noise and dark current due to their unique
detection mechanism compared to conventional detectors such as CCDs[1, 10, 4],
empirical evidence has shown that they do indeed still measure photon-like
events when they are not illuminated that may be triggered by noise sources
such as those noted in Section 3.2. Since the MKID Analog Readout saves photon
pulse timestream data (as in Figures 1 and 2) it is possible to explore the
nature of these events to determine if these ‘dark counts’ look the same as
‘true’ photon events, or if they are demonstrably different. By measuring
several thousand ‘dark’ photons this way we aim to assess them and determine
if we can assign them any explainable origin.

###  5.2 Data collection and reduction

A second set of data was collected in addition that from the MKID Digital
Readout while the MKID was unilluminated where an MKID pixel was read out
using an analog readout system designed for low noise, single-pixel
characterization.

We chose a single MKID resonator that was also read out using the digital
readout that had above-average resolving powers at all calibration energies.
In this configuration there is only one MKID pixel read out, so we are no
longer able to leverage the bulk properties of the MKID array for cosmic ray
rejection. However, the analog readout saves the phase timestream of the
resonator surrounding each photon event which allows inspection of each pulse
(see Figures 1 and 2) to determine whether it is characteristic of a ‘real’
photon or whether noise caused the trigger.

In contrast to the digital readout which continuously takes data until the
user decides to stop, the analog readout accepts the number of desired photon
counts to measure before stopping. In this case the quiescent count rate of
photon counts in the dark was first measured and found to be $\sim$0.03 Hz,
which likely consists of predominantly cosmic ray events. With this in mind we
chose to register 8000 photon counts and expected this should take roughly 3
days. The primary goal of this investigation is to see if the photons which
are being triggered on look ‘real’ or if they look like noise, although it is
also possible to ascertain a dark count rate.

The analog readout system saves its data in a slightly different structure
than the digital readout of Section 4. The digital readout saves time-tagged
lists of photons along with the pixel location and height of the photon event
but due to computational constraints do not save further information about the
photons. In contrast, the analog readout operates in a way so that the recent
phase data from each MKID pixel being read out is kept in memory so that when
a photon is measured, the readout system may save a timestream of that phase
data from the pixel from the time surrounding the photon event. An example of
a phase timestream saved by the analog readout is shown in Figure 1. The
duration and sample rate of this phase timestream are both parameters that can
be tuned by the user. In this experiment the resonator’s phase was sampled at
0.8 MHz and each phase timestream saved the 5000 $\mu$s surrounding the photon
event (2500 $\mu$s before and after).

Analogously to the MKID Science Data pipeline, we first reject any photons
which are outside of the calibrated bandpass (i.e. the peak of the phase is
too high or too low) as well as any timestreams that contain more than 1
photon hit. The second criterion is the closest proxy we have to a time-
coincidence veto in lieu of using bulk statistics from many pixels. This
leaves 1118 photons, $\sim 14\%$ of the total observed counts. In comparison,
the cuts from the digital readout left us with $\sim 10.7\%$ of the total
observed counts (94262 remained of the initial 880855 from the analyzed
pixels). The percentage for the analog readout is much due to both the lower
noise from the readout system resulting in fewer counts below the calibrated
region (i.e. triggers on noise) as well as the inability to make simultaneity
cuts on cosmic ray events in the analog case.

###  5.3 Analysis

By binning photons using the same energy bins as in the MKID Digital readout
we are able to compare the improvement that is gained when reading out a
single pixel using significantly less noisy readout electronics. The count
rates within the calibrated bandpass are shown in Table 3. As in the digital
readout analysis the errors on photon counts and ultimate count rates are
calculated using Poisson statistics.

| Analog Readout | Digital Readout (No Reduction)
---|---|---
Bin Center | Count Rate ($\times 10^{-3}$) | Total Counts | Count Rate ($\times 10^{-3}$) | Total Counts
(eV) | (photons/pixel/s) | (photons) | (photons/pixel/s) | (photons)
1.005 | 1.8$\pm$0.1 | 189$\pm$14 | 5.0$\pm$0.1 | 436.7$\pm$0.9
1.123 | 1.4$\pm$0.1 | 148$\pm$12 | 3.83$\pm$0.09 | 332.2$\pm$0.8
1.240 | 1.4$\pm$0.1 | 145$\pm$12 | 3.17$\pm$0.08 | 275.2$\pm$0.7
1.358 | 1.4$\pm$0.1 | 141$\pm$12 | 2.73$\pm$0.07 | 237.1$\pm$0.6
1.476 | 1.2$\pm$0.1 | 121$\pm$11 | 2.44$\pm$0.07 | 211.6$\pm$0.6
Table 3: Comparison of the total counts in the calibrated bandpass between
0.946-1.534 eV (1310-808 nm) using the MKID analog readout scheme to the MKID
digital readout system.

####  5.3.1 Photon rise and fall times

As previously discussed, the analog readout system saves photon timestream
data (such as those shown in Figures 1 and 2). This allows us to examine the
characteristic rise and fall times of the MKID pixel’s phase when a photon
event is triggered. For a baseline measurement an 808 nm (0.946 eV) laser and
a 1310 nm (1.534 eV) laser are each shined on the pixel until it has
registered 20,000 photon events.

A photon event is characterized by a fast exponential rise time in the
measured phase as a photon strikes the detector, depositing its energy and
breaking Cooper pairs into quasiparticles followed by a slower exponential
tail as the quasiparticles recombine. To fit the rise and the fall times for a
given photon event, the timestream is first split into a rising portion from
the start of the timestream to the peak and a falling portion from the peak to
the end. The rise and fall times are then each calculated from their
respective sections of the timestream by fitting an exponential of the form

$\phi=Ae^{-(t/b)}+c$ (4)

where $\phi$ is the measured phase, $t$ is the corresponding time within the
photon timestream, $b$ is the time constant (which we call the rise or the
fall time depending on which part of the event we are fitting), and $A$ and
$c$ are constants to account for any offset or scaling differences between
pulses.

Figure 8: Comparison of the rise times (top) and fall times (bottom) as a
function of photon wavelength between dark counts - shown in black - and
photons from 808 and 1310 nm lasers - shown in blue and red, respectively. The
error bars for each point are the 1$\sigma$-errors for each measurement. In
the top panel it can be clearly seen that the rise times of dark count photons
and laser photons overlap significantly. In the bottom panel, the fall times
for the dark count events do not overlap the photon events from the lasers.

Figure 9: Comparing the calculated rise and fall times of photons of the same
energy measured by the MKID resonator. (Top) An 808 nm photon measured with an
808 nm laser shining on the pixel. (Bottom) An 808 nm photon measured when
there is no light incident on the pixel. This corroborates the significant
difference in fall times shown in Figure 8.

We fit the rise and fall times for all of the photons from both the 808 and
1310 nm lasers as well as the dark-count photons measured by the pixel while
it was unilluminated. Figure 8 shows how the rise and fall times of the dark
counts compare to those of known photons from the two lasers as a function of
wavelength.

It can be clearly seen that the 1$\sigma$ (68$\%$) confidence intervals of the
rise times of the dark count photons overlaps with the 1$\sigma$ CI of the
rise times of the 808 and 1310 nm photons. This is relatively unsurprising as
phase spikes so sharply when energy is deposited into the pixel that it is
effectively instantaneous even with the microsecond timing resolution.

On the other hand, when the fall times of the dark counts are compared to
those of the photons from the lasers one can see a stark distinction between
the two distributions. The fall times from the laser photons are significantly
faster than those of the dark counts for photons of similar energy - and
therefore similar phase response. In theory the phase decay back to its
quiescent value is governed by the quasiparticle recombination time, an effect
solely due to properties of the superconducting material. Since the phase
response is proportional to the number of quasiparticles generated in the
material, photons that cause similar phase responses should have similar fall
times since a roughly equivalent number of quasiparticles need to combine back
into Cooper pairs. We can see in the bottom panel of Figure 8 that this is not
the case in this experiment. Figure 9 shows an example of each of these
photons. In the top panel, a photon that was measured when a laser was
incident on the pixel can be seen with a measured fall time of 32.07 $\mu$s.
Below that a second photon of the same energy can be seen, this time from when
there was no light incident on the array. The fall time in this case is
measured to be 79.89 $\mu$s, consistent with the difference in the
distribution of the fall times from each population.

Although the dark count photons that remain after all previous cuts
qualitatively appear similar to the photons measured when lasers were being
shined on the MKIDs, there is a quantitative difference in the two
populations. The photons that are measured when a laser is being shined on the
MKID show signficantly shorter fall times than dark count photons measured by
the same pixel. The explanation for this behavior stems from the sources of
the different families of photons and how they deposit energy into and are
subsequently measured by the pixel. The rise and fall times of a photon event
correlates with how quickly energy is deposited into the resonator and how
long it takes for that energy to dissipate and allow the resonator return to
its unexcited state. When energy is deposited into the resonator over a short
time it will rapidly break many Cooper pairs which then begin to recombine
right away, allowing the resonator to return to its unexcited state in a short
time. If energy is deposited into the resonator over a longer time scale it
will still break many Cooper pairs quickly but as the energy remains in the
system before dissipating it will prevent those Cooper pairs from combining as
fast which leads to a longer fall time back to quiescence. Real photons from a
laser or other light source fall into the first category; the photon is
absorbed and all of its energy is immediately deposited into the MKID, leading
to a fast rise and fall time. Dark counts from sources such as cosmic rays are
in the second family. Photon events triggered by a cosmic ray occur when the
energy from the incident energetic particle is down-converted into a cloud of
phonons that then spread the absorbed energy through the array substrate. As
these phonons move past MKID pixels they deposit some of that energy into the
pixel over the time it takes for them to move through the resonator. This
means that dark count photons from cosmic ray events are non-instantaneous
process and will consequently have longer fall times.

A separate potential explanation for the discrepancy in fall times between the
laser photons and dark count photons is that while illuminating the array with
a laser the photon flux is significantly higher than when the laser is off.
This higher photon flux may lead to slight heating of the thin film of the
MKID array. In turn, this would lead to shorter quasiparticle recombination
times and ultimately shorter fall times for each photon event.

There is currently insufficient evidence to conclusively say that these are
two completely different populations of photons due to the current gap in
understanding of the noise sources in the MKIDs and inability to
simultaneously read out an array using both the digital and analog readouts to
correlate cosmic ray events (digital) to single photon traces (analog) so we
cannot calibrate these photons out based solely on the difference in their
fall times than is expected. If the assumption is made that these counts do
come from cosmic rays and a calibration cut is made, the number of counts
would decrease significantly from 121 to 10 photons in the high energy bin and
from 189 to 27 photons in the lowest energy bin, with each bin seeing a
reduction by about a factor of 8. Across the full bandpass the count rate
would fall from $(7.1\pm 0.3)\times 10^{-3}$ photons/pixel/s to $(9.3\pm
0.9)\times 10^{-4}$ photons/pixel/s. In wavelength space this corresponds to
$(1.42\pm 0.05)\times 10^{-5}$ photons/pixel/s/nm and $(1.9\pm 0.2)\times
10^{-6}$ photons/pixel/s/nm for the ‘uncalibrated’ and ‘calibrated’ rates,
respectively. The final value with long-fall time events removed is just
slightly below the $(2.77\pm 0.02)\times 10^{-6}$ photons/pixel/s/nm from the
digital readout in Section 4.3.

##  6 Discussion and conclusions

In this paper we show that across the calibrated MKID bandpass from 0.946 to
1.534 eV (1310-808 nm), the count rate seen by the detectors in a large format
array is $(3.14\pm 0.01)\times 10^{-3}$ photons/pixel/s/eV or $(3.68\pm
0.01)\times 10^{-6}$ photons/pixel/s/nm. It as also demonstrated that by using
a relatively light calibration cut for cosmic ray events we are able to reduce
the number of spurious photon events by nearly a factor 10 of while removing
less than 1$\%$ of the duration of the data collection.

Using the MKID analog readout system and recording the shape of photon pulses
in a single pixel we first show that the count rate across the calibrated
bandpass is $(1.42\pm 0.05)\times 10^{-5}$ photons/pixel/s/nm without any
further data cleaning steps, demonstrating that a quieter system does lead to
lower count rates in an unilluminated MKID device.

While investigating the shape of the photon pulses using the analog readout it
was found that the exponential tail of the dark counts corresponds to a
significantly longer fall time than from photons generated from a laser at the
same wavelengths. The long fall times are indicative of energy taking a long
time to dissipate from the resonators which points to events causing these
triggers happening in the substrate rather than the pixels themselves. An
example of a known event that takes place in the substrate and causes
contaminating photon events is a cosmic ray hitting the MKID array. Making
MKID pixels atop membranes is an ongoing field of research that offers a
straightforward way to to minimize the potential for substrate absorptions to
cause contaminating photon events. Because of the added complexity of making
MKIDs on membranes in addition to the existing difficulty in fabrication this
is not feasible for the large-format MKID arrays currently in use. However, in
future experiments requiring much fewer (1 to $\sim$100) MKIDs, making them on
membranes may be a reasonable path forward that will help mitigate
contamination from substrate absorptions. Additionally, the current generation
of MKID readout hardware does not allow for side-by-side simultaneous readout
of many pixels while still capturing the photon phase timestreams and so there
is no way to determine if these long fall time photons are coexistent with
cosmic ray events at present. The next 3rd Generation MKID Digital Readout is
currently under development [18] and promises to allow both capabilities at
the same time. Further investigation of the source of these dark photons and
if they are in fact generated cosmic rays will be explored in future work with
the 3rd Generation MKID Readout. We note here that if future work does find
that these long fall time photons are from contaminating sources it offers a
straightforward way to calibrate them out of MKID datasets.

For a future dark matter detector experiment that would use a 100 pixel MKID
array with 10 nm energy bins, the maximum dark count rate in the detector
would be $\sim(3.68\pm 0.01)\times 10^{-3}$ photons/s if the current style
arrays and generation of MKID digital readout were used. With this said, any
future MKID dark matter direct detection instrument will have several key
upgrades to mitigate noise in the system. First, a new generation of MKID
readout is currently under development which promises to be a significantly
less noisy system than the one used at present. The continued development of
Traveling Wave Parametric Amplifiers (TWPA [15, 3]) will also significantly
reduce system noise compared to the more commonly used HEMT amplifiers.
Finally, this instrument itself will use an array that has anti-reflection
(AR) coating on the MKID devices and will not have optics that allow visible
light to enter the cryostat. Both of these upgrades will prevent more stray
photons from entering the fridge and causing spurious, unattributable counts
on the detector.

##  7 Acknowledgements

NS gratefully acknowledges support from the Heising-Simons Foundation under
grants $\\#$2020-1820 and $\\#$2021-3058, as well as support by NASA under
grant $\\#$80NSSC19K0329 and from the NSF MRI Award $\\#$1625441\. WHC is
grateful for the support from NSTGRO Grant $\\#$80NSSC21K1290.

##  8 Disclosures

The authors declare no conflicts of interest.

Data availability. Data underlying the results presented in this paper are not
publicly available at this time but may be obtained from the authors upon
reasonable request.

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