# Statistical Inference for Coadded Astronomical Images

Mallory Wang
Department of Statistics
University of Michigan
<EMAIL_ADDRESS>
&Ismael Mendoza∗
Department of Physics
University of Michigan
<EMAIL_ADDRESS>
Cheng Wang
Department of Statistics
University of Michigan
<EMAIL_ADDRESS>
&Camille Avestruz
Department of Physics
University of Michigan
<EMAIL_ADDRESS>
&Jeffrey Regier
Department of Statistics
University of Michigan
<EMAIL_ADDRESS>These authors contributed equally.

###### Abstract

Coadded astronomical images are created by stacking multiple single-exposure
images. Because coadded images are smaller in terms of data size than the
single-exposure images they summarize, loading and processing them is less
computationally expensive. However, image coaddition introduces additional
dependence among pixels, which complicates principled statistical analysis of
them. We present a principled Bayesian approach for performing light source
parameter inference with coadded astronomical images. Our method implicitly
marginalizes over the single-exposure pixel intensities that contribute to the
coadded images, giving it the computational efficiency necessary to scale to
next-generation astronomical surveys. As a proof of concept, we show that our
method for estimating the locations and fluxes of stars using simulated coadds
outperforms a method trained on single-exposure images.

##  1 Introduction

The next generation of astronomical surveys, including the Rubin Observatory’s
Legacy Survey of Space and Time (LSST), will produce massive quantities of
image data. Measurements of large structures in our universe based on this
data will allow us to constrain cosmological parameters to unprecedented
levels of precision [1]. These analyses rely on combining multiple single-
exposure astronomical images into a single coadded image by linearly combining
stacked pixel intensities. Relative to the single-exposure images on which
they are based, coadds are nearly three orders of magnitude smaller in terms
of data size and have greater image depth [2]. As a result, coadded images are
much easier to analyze: they are faster to load and have higher signal-to-
noise ratios.

However, coadded images are more difficult to interpret in a statistically
principled way than single-exposure images. The single-exposure images
contributing to a coadd vary in point spread function (PSF) and require
resampling to be aligned to a common grid [3, 4]. Resampling and PSF
homogenization create additional covariance between nearby pixels,
complicating statistical inference on coadds; whereas in the raw images, pixel
intensities are well modeled as independent conditional on the latent
properties of nearby stars, galaxies, and the background, in coadds they are
not.

Bayesian Light Source Separator (BLISS) is a scalable probabilistic method for
detecting and cataloging astronomical objects using single-exposure images
[5]. In the current work, we extend the BLISS model to create a fully
generative model of coadded images (Section 2). The true coaddition process is
deterministic and well understood, so modeling it requires few assumptions. In
single-exposure images, BLISS models pixel intensities as conditionally
independent given the latent properties of imaged light sources; each pixel
has an observed pixel intensity sampled from a Poisson distribution with the
underlying intensity, unique to that pixel, as its rate. This approach breaks
down for coadds, in which pixel intensities are not independent given the
latent properties of imaged sources. To account for coadds, we add a unique
latent variable for each pixel in a raw image that contributes to the observed
coadd.

We leverage forward amortized variational inference (FAVI) [6], a
computationally efficient approach that implicitly marginalizes over the new
latent variables corresponding to the pixel values from single-exposure
observations (Section 3). These new latent variables are nuisance variables;
we wish to infer only the properties of imaged light sources. FAVI effectively
ignores large numbers of nuisance latent variables in its optimization
routine. This approach is thus a computationally efficient way of performing
principled Bayesian inference on coadded images.

We compare results of four trained BLISS encoders with varying numbers of
coadded single-exposures. We find that our method applied to coadds outperform
the method applied to single-exposures, in terms of detection and flux metrics
(Section 4). Our results illustrate the feasibility of performing principled
end-to-end Bayesian inference on image coadds for downstream analyses. The
source code for these studies is available at https://github.com/prob-
ml/bliss.

##  2 Statistical Model

We first describe the generative model for single-exposure images. We generate
single-exposures of size $90\times 90$ following [7]. We generate simulated
images of stars using a constant (but realistic) PSF, denoted $\Pi$, from SDSS
[8]. We characterize stars with two parameters: location $\ell$ and flux $f$.
Our single-exposures model an SDSS r-band exposure with only stars.

We sample star parameters from the prior as follows. First, we sample the
number of stars $S$ in the image from a Poisson distribution with mean rate
$\lambda=3$. We uniformly sample independent centroids for each source
$s=1,2,...,S$ within the central $40\times 40$ pixel square of the image.
Finally, the flux follows a truncated power law distribution:
$f_{s}\sim\text{Pareto}(f_{\rm min},f_{\rm max},\alpha)$, where $f_{\rm
min}=622$ counts, $f_{\rm max}=10^{6}$ counts, and $\alpha=0.47$. We denote
the catalog $z=\\{\ell_{s},f_{s}\\}_{s=1}^{S}$ as the collection of the
parameters of all the stars in that image.

To generate a coadd $y$, we first need to create the set of $d$ aligned
single-exposures $\boldsymbol{x}=\\{x^{(i)}\\}_{i=1}^{d}$ that are stacked for
a given coadd. To create this set, we repeat the single-exposure procedure $d$
times with the same catalog $z$, but we vary (dither) the location of every
star in a given single-exposure by the same random sub-pixel shift between
$-0.5$ and $0.5$ in both horizontal and vertical directions. We independently
sample this location noise for each exposure. We then align the $d$ single-
exposures using a reference grid $G_{0}$, which corresponds to the not-
dithered reference single-exposure $x_{0}$. We bilinearly interpolate the
dithered single-exposures at the grid points $G_{0}$, then crop the border of
all exposures, resulting in $88\times 88$ images. The result is the set of
aligned single-exposures $\boldsymbol{x}=\\{x^{(i)}\\}_{i=1}^{d}$ all
evaluated on a common reference grid $G_{0}$. The top panel of Figure 1
illustrates our alignment and interpolation procedure. Note, the interpolation
step is a weighted average between adjacent pixels given the sub-pixel dither.
Critically, the alignment procedure induces a correlation between adjacent
pixels and has a smoothing effect: each aligned exposure no longer follows a
Poisson noise model.

In our model, the coadd is a weighted sum of the single-exposures images:
$y=\sum_{i=1}^{d}w^{\prime}_{i}x^{(i)}$ where
$w^{\prime}_{i}=w_{i}/\sum_{j=1}^{d}w_{j}$. We use an inverse variance
weighting scheme, which results in an optimal signal-to-noise ratio for
detecting objects in the coadd [9]. Specifically, the weights are
$w_{i}=(b+u^{(i)})^{-1}$, where $u^{(i)}$ is the signal of the single exposure
$x^{(i)}$ and $b$ is the constant background intensity ($b=865$ counts).

The bottom panel of Figure 1 illustrates an example single-exposure and
corresponding coadds of $d=5,10,25$ from left to right. By increasing $d$, we
can increase the signal-to-noise ratio (SNR) of light sources; objects
undetectable in a single-exposure may be more visually obvious in a coadd.

Figure 1:  Top: Cartoon outlining grid interpolation for alignment. Bottom:
Comparison of a single-exposure (leftmost panel) with coadd images comprised
by an increasing number of single-exposures $d=5,10,25$. All images contain
stars with the same locations and fluxes and have dimensions $88\times 88$
pixels. For more details on the generation process see Section 2.

##  3 Posterior Inference

We wish to approximate the posterior distribution $p(z\mid y)$ of the catalog
$z$ given the observed coadd image $y$. However, the coadd image is
potentially made up of many single-exposure images
$\boldsymbol{x}=\\{x^{(i)}\\}_{i=1}^{d}$. In principle, a fully Bayesian
approach would need to model both the catalog $z$ and the random pixel
intensities belonging to all single-exposures $\boldsymbol{x}$ given the
observed coadd $p(z,\boldsymbol{x}\mid y)$, and then marginalize over
$\boldsymbol{x}$ by performing the integral
$\int_{\boldsymbol{x}}p(z,\boldsymbol{x}\mid y)d\boldsymbol{x}$. This integral
is computationally intractable given the very large number of latent variables
that need to be marginalized over (one per pixel of each single-exposure
making up the coadd). For our model, common Bayesian inference techniques such
as MCMC and traditional variational inference would be extremely
computationally expensive to perform because they would require explicitly
modeling large numbers of nuisance latent variables. In general, these
nuisance latent variables cannot be marginalized analytically. Instead, we use
Forward Amortized Variational Inference (FAVI) [6], a recently developed
likelihood-free approach to amortized variational inference that uses the
forward Kullback-Leibler (KL) divergence in a joint-contrastive variational
loss.

Assuming a flexible enough encoder, the global optima of the FAVI loss is
guaranteed to be $p(z\mid y)$ with a FAVI loss that consistently marginalizes
$\boldsymbol{x}$. The FAVI loss implicitly marginalizes over $\boldsymbol{x}$
simply by excluding these variables from the objective, without having to
consider their dependency with the catalog $z$. This allows us to approximate
$p(z\mid y)$ directly with a variational distribution $q(z\mid y)$, and
optimize choosing $q$ based on the FAVI loss as described in detail next.

### Training and Architecture

Our inference routine reuses BLISS [5] code, which already employs the FAVI
loss to infer light source counts, locations, and fluxes in single-exposure
images. In [5], the input to the encoder neural network consists of single-
exposure images, each split into equal-sized tiles. The weights of the neural
network, $\phi$, parametrize a variational distribution $q_{\phi}(z\mid x)$.
The variational distribution approximates the true marginal posterior,
$p(z\mid x)$, on the catalog, $z$, given a single-exposure, $x$. We choose a
variational distribution that factorizes over the tiles. For each tile, the
variational distribution further factorizes over a categorical distribution
for the number of light sources, a logit distribution for the (normalized)
locations, and a log-normal distribution for the fluxes. The encoder outputs
the parameters in each tile, which are used to compute FAVI loss on each tile
and then summed over all tiles of each image. This loss is used to optimize
the neural network weights $\phi$.

To extend BLISS to coadds, we retrain the BLISS encoder uses the training
procedure in [5, 7] to approximate the marginal posterior of the true catalog
given the coadd with a new variational distribution $q_{\phi}(z\mid y)$. The
only modification required is to use coadds as images for training instead of
single-exposures. We train four different encoders with different numbers of
$d$ coadded single-exposures: 1, 5, 10, 25. The encoder trained with one
exposure is just the original BLISS encoder trained on single-exposures with
no dither and no interpolation. For each of these encoders, we fitted a
variational distribution that outputs counts, locations, and fluxes of stars.

The neural network in the encoder consists of a combination of convolutional,
batch-norm, and dropout layers using ReLUs as an activation. We use PyTorch
[10] and the Adam optimizer with a learning rate of $10^{-4}$. For
optimization, we use an NVIDIA GeForce RTX 2080 Ti GPU. For both the single-
exposure encoder and the coadd encoders we trained for 30 epochs, where each
epoch consisted of a fixed set of 10000 images of size $88\times 88$. Each
encoder type was trained on the same dataset five times with different random
seeds in order to estimate epistemic uncertainty.

##  4 Validation Metrics and Results

Figure 2 shows the results of the trained BLISS encoders with increasing
numbers of single exposures ($d=1,5,10,25$), evaluated on test datasets with
their respective number of coadded single-exposures. The results we report are
based on the mode of the variational distribution for star locations and
fluxes.

For objects brighter than $21.5$ magnitude,111As a reference, the brightest
stars we considered in our results with magnitude $20$ correspond to an
(isolated) SNR of around $45$ on a single exposure and $100$ for the coadd
with $d=25$. On the other hand, the dimmest stars with magnitude $23$ have an
SNR of around $3$ for the single exposure and $7.5$ on the coadd with $d=25$.
the precision and recall values are similar for the single-image and coadded-
image results. However, as objects become dimmer, the performance of the
single-exposure encoder rapidly decreases, with the recall reaching $0.25$ and
precision reaching $0.65$ in our dimmest objects of about $23$ magnitude. For
the coadd encoders, the precision remains mostly constant for all magnitudes.
The recall for coadd encoders starts decreasing past magnitude $21.5$, but
stays above $0.70$ for even the dimmest objects. The recall across magnitudes
in Figure 2 follows the expected ordering with higher number of single-
exposure in coadds.222We noticed a saturation in detection performance for
coadd encoders with $d>25$. The difference in recall and precision became
statistically insignificant past $d=25$.

Finally, the residual flux median are nearly zero for all magnitudes, but the
residual fluxes from the single-exposures have larger scatter than the coadd
ones. This suggests higher confidence predictions on flux by the coadd
encoders, which makes sense given the higher SNR in their images. Moreover,
the single-exposure encoder predicts a residual median that exceeds $20\%$ in
the dimmest magnitude bin. All coadd encoders output an absolute median
residual below $10\%$ and have lower scatter than that of the single-exposure.
Thus, the coadd method outperforms the single-exposure method in predicting
flux.

Figure 2:  Left and center: Precision and recall of source identification,
computed by matching predictions with true locations within one pixel. Bands
indicate 25–75 percentiles of the bootstrap samples across the five training
seeds. Right: Median fractional residual of the predicted flux of each matched
light source, with bands indicating 25–75 percentile values of the residuals.

##  5 Conclusion

We present a novel fully Bayesian approach for performing light source
parameter inference on coadded images. Our method implicitly marginalizes over
the single-exposure pixel intensities that contribute to the coadd and is
computationally efficient. We show preliminary results of our model in
estimating star locations and fluxes using simulated coadds containing stars.
Our method on coadds outperforms that of the single-exposure in both detection
and flux metrics. Thus, it has the potential to scale to next-generation
astronomical surveys and provide calibrated uncertainties for the measured
properties of potentially billions of detected light sources. Future work will
be devoted to extending the model to more realistic simulated coadds.
Additionally, we will propagate uncertainties to downstream analyses, such as
in photometric redshift estimation.

Broader Impact: Our method has potential implications for quantifying
uncertainty in the context of classification and detection in noisy images.
Given the differences between astronomical images and real-world images, we do
not expect our method to be misused for purposes like surveillance. Our
approach can have a positive impact on the growing environmental cost of
computing. Whereas other solutions such as MCMC may require more time and
computational power, our approach has the potential to be more efficient in
both aspects.

## Acknowledgments and Disclosure of Funding

IM and CA acknowledge support from DOE grant DE-SC009193.

## References

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# Onset of nonequilibrium in a driven Anderson insulator

Z. Ovadyahu  Racah Institute of Physics, The Hebrew University, Jerusalem
9190401, Israel

###### Abstract

The onset of nonequilibrium in a driven Anderson-insulator is identified by
monitoring the system with two-thermometers. Features of nonequilibrium appear
at surprisingly weak drive intensity demonstrating, among other things, that
conductivity may not be a reliable thermometer for ensuring linear-response
conditions. In addition, the spectral contents of the applied field could be
more important to take the system out of equilibrium than its absorbed power.
Ensuing hot-electron transport effects and the nontrivial role phonons play in
driven quantum systems are pointed out.

††preprint:

Below 1K, you know the temperature if you have one thermometer; when you have
two, you don’t. This adage is recalled whenever a sample resistance fails to
follow the expected dependence on temperature given by the fridge–installed
thermometer. Such an event is often related to an influx of energy causing a
lack of detailed-balance with the bath. The sample may still be in a steady-
state but not in thermal-equilibrium. Considerable attention has been given to
nonequilibrium states, created by a variety of means, to look for novel
phenomena 1 ; 2 ; 3 ; 4 ; 5 ; 6 . A non-equilibrium steady-state (NESS) is a
common occurrence in low temperature transport experiments making it a natural
testbed for these studies.

Monte-Carlo simulations of driven disordered systems supported the intuitive
notion that the electrons may be assigned an effective temperature
T${}_{\text{eff}}$ distinctly different from the phonon temperature 7 ; 8 .
The effective temperature of the electrons was defined by fitting their energy
distribution to the Fermi-Dirac expression 7 ; 8 . The relevance of an
effective temperature under drive by external fields is one of the issues
tested experimentally here.

In this note, we outline a method to detect the onset of nonequilibrium in a
driven electronic system by comparing the readings of two thermometers. These
are attached to the electronic conductance of an interacting Anderson-
insulator, aka the electron glass 9 ; 10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16 ; 17 .
Once driven by external fields, both thermometers read higher temperatures
than that of the bath. As the drive power increases from the lowest level of
the experiment, they exhibit different T${}_{\text{eff}}$’s signalling a
breakaway from thermal equilibrium. The difference between the two
thermometers readings, $\Delta$T${}_{\text{eff}}$ increases monotonously with
the drive intensity. Significantly, $\Delta$T${}_{\text{eff}}$ turns out to be
sensitive to the spectral contents of the energy supplied by the drive, not
just its magnitude. The results shed further light on the glassy nature of the
Anderson insulating phase, and in particular, on the reason for its sluggish
dynamics vis-a-vis the much faster transitions involved in the conductance.

Samples used in this study were 20nm thick films of amorphous indium oxide
(In${}_{\text{x}}$O). These were made by e-gun evaporation of 99.999% pure
In${}_{\text{2}}$O${}_{\text{3}}$ onto room-temperature Si wafers in a partial
pressure of 3x10${}^{\text{-4}}$mBar of O${}_{\text{2}}$ and a rate of
0.3$\pm$0.1Å/s. The Si wafers (boron-doped with bulk resistivity
$\rho\leq$2x10${}^{\text{-3}}\Omega$cm) were employed as the gate-electrode in
the field-effect and experiments. The samples were deposited on a
SiO${}_{\text{2}}$ layer (2$\mu$m thick) that was thermally-grown on these
wafers and acted as the spacer between the sample and the conducting Si:B
substrate. The carrier-concentration N of these samples, measured by the Hall-
Effect at room-temperatures, yielded carrier concentration
N$\approx$(1$\pm$0.4)x10${}^{\text{19}}$cm${}^{\text{-3}}$.There are to date
six Anderson insulators that may be used for these experiments 18 . The
motivation for choosing this version of In${}_{\text{x}}$O was its highly
pronounced memory-dip, which is used here as a sensitive thermometer.

Conductivity of the samples was measured using a two-terminal ac technique
employing a 1211-ITHACO current preamplifier and a PAR-124A lock-in amplifier.
Measurements were performed with the samples immersed in liquid helium at
T$\approx$4.1K held by a 100 liters storage-dewar. This allowed up to two
months measurements on a given sample while keeping it cold. These conditions
are essential for measurements where extended times of relaxation processes
are required at a constant temperature, especially when running multiple
excitation-relaxation experiments on a given sample. The gate-sample voltage
(referred to as V${}_{\text{g}}$ in this work) in the field-effect
measurements was controlled by the potential difference across a 10$\mu$F
capacitor charged with a constant current. The rate of change of
V${}_{\text{g}}$ is determined by the value of this current. Except when
otherwise noted, the ac voltage bias used in conductivity measurements was
small enough to ensure near-ohmic conditions. Exciting the system by infrared
radiation was accomplished by a light-emitting AlGaAs diode operating at
$\approx$0.85$\pm$0.05$\mu$m. It was placed $\approx$15mm from and facing the
sample surface. The thermometers used for the study are two distinct
properties of electron glasses; the conductance G, and its sensitivity to a
change of the carrier-concentration N. The latter was affected by changing the
voltage V${}_{\text{g}}$ between the sample and a nearby gate electrode. A
typical field-effect measurement using this MOSFET configuration is shown in
Fig. 1.

Figure 1: Fig.1- Conductance vs. gate-voltage V${}_{\text{g}}$ for a 20nm
thick In${}_{\text{x}}$O film with 1mmx1mm lateral size. Data were taken with
a sweep-rate $\partial$V${}_{\text{g}}$/$\partial$t of 1.2V/s at a bath
temperature T=4.11K. The values of the two parameters used in the text,
G${}_{\text{0}}$ and G${}_{\text{1}}$, are taken as the intercepts of the
arrows with the ordinate. The ”bump” at V${}_{\text{g}}\approx$6V is a
consequence of charge ordering as explained in 19 .

The figure also illustrates how the parameters G${}_{\text{0}}$ and
G${}_{\text{1}}$, used for determining the temperature of the two thermometers
are defined. The first thermometer reading is the steady-state conductance
G${}_{\text{0}}$=G(V${}_{\text{g}}$=0). The second thermometer reading is
$\eta$=(G${}_{\text{1}}$-G${}_{\text{0}}$)/G${}_{\text{0}}$ where
G${}_{\text{1}}$ is determined by the construction in Fig. 1. The same sweep-
rate $\partial$V${}_{\text{g}}$/$\partial$t =1.2V/s was used in all
G(V${}_{\text{g}}$) traces in this work. Note that the need to complete the
G(V${}_{\text{g}}$) trace makes the second thermometer ‘slower’ than the
first, a difference that plays a role in their response. Both G${}_{\text{0}}$
and $\eta$ turn out to be strongly temperature-dependent making them sensitive
‘secondary’ thermometers. Applying them for temperature measurements requires
calibration charts for each. These are shown in Fig. 2 and Fig. 3 for
G${}_{\text{0}}$(T) and $\eta$(T) respectively.

Figure 2: Pertinent transport data taken for the sample. (a): The relative
change of conductance as function of the applied electric field. (b): The
conductance vs. bath-temperature of the sample measured by two-terminal ac
technique using a bias voltage in the linear-response regime at a frequency
f=73Hz. At each point, the sample was allowed to relax for 1-2 hours to ensure
steady-state conditions. The dashed line is a fit to:
G${}_{\text{0}}$(T)$\propto\exp$[-(T${}_{\text{0}}$/T)${}^{\text{1/3}}$] with
T${}_{\text{0}}\simeq$5800K. Figure 3: Conductance vs. gate-voltage
G(V${}_{\text{g}}$) for the In${}_{\text{x}}$O film. Curves were taken at
indicated temperatures with the same sweep-rate:
$\partial$V${}_{\text{g}}$/$\partial$t=1.2V/s.

Figure 4 shows a set of G(V${}_{\text{g}}$) curves taken by using non-Ohmic
fields to measure both, the sample conductance and its associated field
effect. All other aspects of the measurements protocol were identical to those
used in obtaining the data set in Fig. 3.

Figure 4: The same as in Fig. 3 except that the G(V${}_{\text{g}}$) curves
were taken under various electric fields.

Evidently, the two sets shown in Fig. 3 and Fig. 4 bear strong resemblance to
one another. The conditions under which these data were collected however, are
quite different; the set in Fig. 3 was taken at equilibrium while that in Fig.
4 was taken as a NESS. The similarity in terms of how the G(V${}_{\text{g}}$)
curves evolve with either temperature or field may tempt one to assign a
T${}_{\text{eff}}$ for a given G(F). At the same time, an effective
temperature T${}_{\text{eff}}$(F) may be independently inferred from the data
sets in Fig. 2 based on G${}_{\text{0}}$. These two protocols for assigning a
T${}_{\text{eff}}$(F) use the same set of measurements while focusing on
different parts of the G(V${}_{\text{g}}$,F) data. They yield, however, quite
different values for T${}_{\text{eff}}$ as is shown in Fig. 5 and Fig. 6
below.

Fig. 5 compares a set of $\eta^{\prime}$s measured at equilibrium temperatures
with the set measured under NESS conditions produced by applying different
fields. The latter is plotted versus T${}_{\text{eff}}$ calibrated against the
data in Fig. 2b on the basis of G${}_{\text{0}}$ values. For brevity sake, we
shall refer to this set of T${}_{\text{eff}}$(F) as that of thermometer #1.
T${}_{\text{eff}}$(F) for the thermometer #2 is then obtained by finding the
temperature on the $\eta$(T) curve that fulfills:
$\eta$(T${}_{\text{eff}}$)=$\eta$(T) as illustrated in Fig. 5 for a specific
$\eta$ value (red arrow). This allows a direct comparison between the
T${}_{\text{eff}}$(F) of the two thermometers depicted in Fig. 6.

It is not surprising that thermometer #1 agrees with thermometer #2 only when
F is vanishingly small. It is however remarkable that the largest relative
discrepancy between the thermometers is where the system is just off the
linear response regime in terms of G${}_{\text{0}}$(F). This observation
demonstrates that T${}_{\text{eff}}$ may be an ill-defined concept in a non-
equilibrium steady-state, even under a fairly weak drive. Such hot-electron
effects 19  are commonly manifested in the transport studies of semiconductors
driven by non-Ohmic fields. The system conductance under these conditions may
also be influenced by electron-electron correlations.

Figure 5: Contrasting the effective temperatures yielded by the two
thermometers under applied fields. Data for T${}_{\text{eff}}$ for thermometer
#1 (squares, and line as guide to the eye). Each $\eta$ taken from Fig. 4 is
assigned a T${}_{\text{eff}}$ by using the data in Fig. 2 for G(T) and G(F)
for the sample. Mapping T${}_{\text{eff}}$ for thermometer #2 is accomplished
by matching the value of the measured $\eta$ with the equilibrium values for
$\eta$(T) taken from Fig. 3. The dashed line is a fit to:
$\eta\propto\exp$[-T/T*] with T*=1.34. Figure 6: A direct comparison of the
effective temperatures for thermometers #1 and #2 based on data from Fig. 5
(see text).

The difference between the two thermometers is probably even larger than
conveyed by Fig. 6. Note that T${}_{\text{eff}}$ for thermometer #1 is
determined here solely on the basis of the measured $\Delta$G(F). However,
$\Delta$G(F) is composed of two contributions. The first is due to field-
assisted tunneling 20 ; 21 ; 22 ; 23 , the second results from Joule-heating
24 ; 25 ; 26 . The T${}_{\text{eff}}$ used in Fig. 6 is therefore larger than
the value based on assuming just heating. This complication, in different
forms, is inherent to resistance-based thermometry commonly used in
experiments on disordered insulators.

Moreover, when the system is driven into a NESS by exposure to IR illumination
the discrepancy between the two T${}_{\text{eff}}$’s turns out to be
significantly larger than under non-Ohmic fields. Figure 7a shows the raw data
for the equilibrium G(V${}_{\text{g}}$) compared with a trace taken under weak
IR illumination.

Figure 7: Conductance versus gate-voltage G(V${}_{\text{g}}$) for the
In${}_{\text{x}}$O sample under two different conditions. (a): Comparing
G(V${}_{\text{g}}$) measured at equilibrium with a trace measured under IR
illumination. The IR source (operating at 0.2$\mu$A while biased at 1.003V)
has been on for 75 minutes before the trace was taken to ensure steady-state
conditions. (b): G(V${}_{\text{g}}$) traces taken from the (equilibrium) data
in Fig. 3 as a pair that exhibit values of $\eta$ similar to these of the
traces in (a).

Note first that the zero-bias conductance of the sample has slightly increased
in the NESS of the sample by $\Delta$G${}_{\text{0}}$, a relative change of
$\approx$2.5%. This is equivalent (using the calibration graph in Fig. 2) to
$\Delta$T$\approx$24mK while (using the data in Fig. 5) the change in $\eta$
is equivalent to $\Delta$T${}_{\text{eff}}\approx$1.1K.

Secondly, To get a comparable shift of $\Delta$T${}_{\text{eff}}$ requires an
applied non-Ohmic field F$\approx$3.7V/mm that, in turn, entails maintaining
input power into the sample of the order of 10${}^{\text{-5}}$W. The total
power used by the light-emitting diode in Fig. 7a is
$\approx$10${}^{\text{-7}}$W 27 . Obviously much less power is required by IR
illumination to reduce $\eta$ than by applying low frequency fields.

Another intriguing feature of the NESS maintained by IR illumination may be
appreciated by comparing the data in Fig. 7a with Fig. 7b. The latter
illustrates characteristic G(V${}_{\text{g}}$) traces of the electron glass
measured in equilibrium. As the bath temperature is raised, or when a larger
non-Ohmic F is applied, the entire G(V${}_{\text{g}}$) trace is shifted to
higher values. By contrast, only G${}_{\text{0}}$ is enhanced when the sample
is exposed to IR while for V${}_{\text{g}}\geq$2V, G(V${}_{\text{g}}$) falls
below the equilibrium value. This feature was consistently observed on three
other samples using IR power $\lesssim$10${}^{\text{-6}}$W.

To all appearances, the difference between the two NESS protocols depends on
the nature of the drive; non-Ohmic field versus IR-excitation.

The natural question is what makes IR illumination so effective in reducing
$\eta$ while barely affecting the system conductance. Additionally, why the
NESS created by non-Ohmic fields requires much larger power to achieve a
similar reduction in $\eta$?

The answer to both questions is related to the way the invested power of a
protocol modifies the phonon distribution D($\varepsilon$) of the system. In
equilibrium, D($\varepsilon$) is given by the Bose-Einstein distribution that,
at cryogenic temperatures, means an abundance of low-frequency phonons and
exponentially rare high-frequency ones.

Applying non-Ohmic fields F mostly adds low-frequency modes to the spectrum.
Down-going transitions associated with hopping under such fields produce
athermal phonons in the spectrum but with energies up to $\approx$eFL, where
L${}_{\text{C}}$=L${}_{\text{C}}$(F,T) is the percolation-radius 28 ; 29 ; 24
; 25 . Typical values for L${}_{\text{C}}\approx$10${}^{\text{-7}}$m at
T$\approx$4K, and fields in the range used in this work, the energy
accumulated from the field will appear as excess of phonons at few degrees
above the bath temperature.

When the system is exposed to the IR source the modification to
D($\varepsilon$) is more profound. The IR initiates a cascade process;
electrons are excited to high-energy, then relax by phonons emission 31 . The
NESS that sets-in, sustains vibrations at phonon-energies of few hundred
degrees 32 . Then, it is the high-energy part of D($\varepsilon$) that is
significantly boosted while the relative change in the low-energy phonon-bank
is small. The greater efficiency of the IR protocol relative to that of non-
Ohmic fields in reducing $\eta$ is a consequence of the excess high-energy
modes it creates. These modes allow transitions that, in equilibrium, would
occur only at temperatures of few hundred degrees where no memory-dip has ever
been observed in any of the half dozen 18  electron-glasses measured todate.

The two thermometers also differ in the typical energy controlling their
response. For thermometer #1 (based on the measured conductance), this is the
optimal hopping-energy 33 ; 34
E${}_{\text{opt}}$=(k${}_{\text{B}}$T)${}^{\text{2/3}}$($\partial$n$/\partial\mu$·d·$\xi^{\text{2}}$)${}^{\text{-1/3}}$.
Here, $\partial$n$/\partial\mu$
$\approx$10${}^{\text{32}}$erg${}^{\text{-1}}$cm${}^{\text{-3}}$ is the
density of states, d is the film thickness, and $\xi\approx$5nm is the
localization length (estimated from G(T) in Fig. 2b). With these values
E${}_{\text{opt}}$ is of order $\approx$1meV at $\approx$5K. The
characteristic energy for thermometer #2 is the disorder energy
W$\gtrsim$0.4eV. This is estimated on the basis of the condition that W is
large enough to Anderson-localize the system and to slow its relaxation such
that the memory-dip is resolved in the field-effect scans 18 .

Hopping conductivity takes place in a current-carrying network (CNN) utilizing
regions of the relatively weak disorder. The CNN encompasses regions of the
highest disorder in the sample 28 ; 29 ; 30 . The large disorder in these
regions is responsible for the slow relaxation of the electron glass 18 . The
two regions 35  communicate mainly via low-energy phonons 18 , and electron-
electron interaction. The range over which interaction is effective however
may depend on available resonances and therefore on quantum coherence in the
medium 36 .

In sum, we have demonstrated that driving an Anderson insulator even slightly
out of its linear response entails a sharp thermometers-conflict suggesting
that thermometry in this regime 37  should be treated with some doubt. The
frequency of the drive turns out to be a pivotal parameter in the thermometers
discrepancy. Explicitly, the difference between the thermometers is
significantly larger when the drive initiates high-frequencies energy quanta
in the medium as occurs in the IR protocol. A specific reason for this was
offered based on the quantum nature of the phonon energy-distribution. Phonons
are an essential ingredient in establishing steady-state conditions in these
NESS experiments. Our results show that they also play a nontrivial role in
nonequilibrium phenomena.

###### Acknowledgements.

The author thanks Preeti Bhandari, Vikas Malik, and Moshe Schechter for
illuminating discussions on their simulation work.

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