# Bayesian information engine that optimally exploits noisy measurements

Tushar K. Saha<EMAIL_ADDRESS>Joseph N. E. Lucero  Department of Chemistry,
Stanford University, Stanford, CA 94305, USA  Jannik Ehrich  David A. Sivak
<EMAIL_ADDRESS>John Bechhoefer<EMAIL_ADDRESS>Department of Physics, Simon
Fraser University, Burnaby, BC, V5A 1S6 Canada

###### Abstract

We have experimentally realized an information engine consisting of an
optically trapped, heavy bead in water. The device raises the trap center
after a favorable “up” thermal fluctuation, thereby increasing the bead’s
average gravitational potential energy. In the presence of measurement noise,
poor feedback decisions degrade its performance; below a critical signal-to-
noise ratio, the engine shows a phase transition and cannot store any
gravitational energy. However, using Bayesian estimates of the bead’s position
to make feedback decisions can extract gravitational energy at all measurement
noise strengths and has maximum performance benefit at the critical signal-to-
noise ratio.

Information engine — Bayesian estimate — Phase transition

Information engines are a new class of engine that use information as fuel to
convert heat from a thermal bath into useful energy. They exploit knowledge of
thermal fluctuations to apply feedback and extract energy from the thermal
bath, while paying costs required by the second law of thermodynamics to
process that information [1]. In the past decade, information engines have
been realized in a wide range of physical systems [2, 3, 4, 5, 6, 7, 8]. For
practical application, it is important to understand how to maximize the
engine’s output [9, 10] and efficiency [11, 12, 13, 10].

One obstacle that degrades the output of an information engine is inaccurate
information about the system that arises from measurement noise. Since
information engines respond to measurements of thermal fluctuations,
measurement noise can lead to wrong feedback decisions. Feedback actions
chosen based on inaccurate measurements reduce the work extracted from the
surrounding thermal bath and can even, at high noise levels, lead to a net
heating of the thermal bath [14].

Previous efforts to account for noisy measurements in information engines have
all used “naive” feedback algorithms based directly on the most recent noisy
measurement [14, 15]. Here we show that such information engines, with
unidirectional ratchets, have a phase transition between working and non-
working regimes: Below a critical level of signal-to-noise ratio for
measurements of the engine state, a “pure” information engine—one that
requires no work input beyond that needed to run the measurement and control
apparatus—is not possible.

Although previous studies noted the degradation of performance due to
measurement noise, they did not attempt to alter the feedback algorithm to
compensate. Yet theoretical studies have indicated that incorporating the
information contained in past measurements via optimal feedback control could
greatly improve the performance of an information engine [15, 16, 17, 18].
Indeed, experiments in other areas of physics have used feedback that
incorporates Bayesian estimators to demonstrate spectacular results, even in
the presence of high measurement noise; significant achievements include
trapping a single fluorescent dye molecule that is freely diffusing in water
[19] and cooling a nanoparticle to the quantum regime of dynamics [20, 21].

In this Letter, we present an experimental realization of an optimal Bayesian
information engine that retains the relevant memory of all past measurements
in a single summary statistic. Using the extra information from past
measurements and correctly compensating for delays in the feedback loop via
predictive estimates, we extract and store significant amounts of energy, even
in the presence of high measurement noise.

Our implementation of the Bayesian filter uses the optimal affine feedback
control algorithm [22], at optimal experimental parameters [10], to maximize
the engine’s rate of gravitational-energy storage. The relevant information
from past observations is used to minimize the uncertainty in the bead’s
position. This Bayesian information engine extracts energy even at low signal-
to-noise ratio (SNR), avoiding the phase transition in the naive information
engine that leads to zero output. Under any conditions, this engine extracts
at least as much work as the naive engine and, indeed, reaches the upper bound
on the performance of Gaussian information engines.

Experimental setup.—Our information engine consists of a 4-µm heavy bead
trapped in a horizontally aligned optical trap. Because of the surrounding
heat bath, the bead’s position fluctuates about an equilibrium average. The
heavy bead is also subject to gravity. The bead’s position is measured with a
sampling time $t_{\mathrm{s}}=20$ µs using the scattered light from a
detection laser.

In the operation of the engine, an observed “up” fluctuation increases the
bead’s gravitational potential energy and can be captured (“rectified”) by a
quick feedback response that shifts up the trap position. The response takes
place after a one-time-step delay $t_{\text{d}}=t_{\mathrm{s}}=20$ µs, with
the shift chosen so that the trap does zero work on the bead. For this “pure”
information engine, the work to run the motor is associated only with the
measuring device and feedback controller, not the engine itself.

The optically trapped bead can be modeled by a spring-mass system [Fig. 1(a)].
The true position of the bead is estimated from a noisy measurement $y$.
Measurement noise is increased by reducing the intensity of the detection
laser beam.

Figure 1(b) shows a naive information engine that directly uses a noisy
measurement $y$ to apply feedback. By contrast, the _Bayesian_ information
engine bases its feedback on the best estimate $\hat{x}$ of the bead’s
position [Fig. 1(c)]. The bead’s position is estimated using a Bayesian filter
that explicitly models the measurement noise and feedback delay 111The delay
is also termed “feedback latency,” which refers explicitly to the sum of
delays in the feedback loop, which include contributions from measurements,
computations, and output to the actuator (the acousto-optic deflector, which
moves the trap center). We use the more familiar informal language in the
text..

The information engine can extract energy at high measurement noise because
feedback decisions based on filtered estimates of the bead’s position are more
likely to ratchet to a “true” upward fluctuation rather than to measurement
noise.

Figure 1: Schematic information engine. (a) Noisy detector measures position
$y$ of the bead, actually located at $x$. Ratchet based on either (b) noisy
position measurement $y$ or (c) Bayesian position estimate $\hat{x}$ (blue
dashed circle).

Equations of motion.—The dynamics of the optically trapped bead obey an
overdamped Langevin equation,

$\displaystyle\gamma\dot{x}=-\kappa\left[x(t)-\lambda(t)\right]-mg+\sqrt{2k_{\mathrm{B}}T\gamma}\
\xi(t)\,,$ (1)

where $x(t)$ is the position at time $t$ of a bead of diameter $d$ and
effective (buoyant) mass $m$ in a trap of stiffness $\kappa$ and center
$\lambda(t)$. The friction coefficient is $\gamma$, and $\xi(t)$ is a Gaussian
random variable with zero mean and covariance
$\expectationvalue{\xi(t)\xi(s)}=\delta(t-s)$. We denote time derivatives by
overdots. Scaling all lengths by the equilibrium position standard deviation
$\sigma\equiv\sqrt{k_{\mathrm{B}}T/\kappa}$ and times by the bead relaxation
time $\tau_{\mathrm{r}}=\gamma/\kappa$ gives the nondimensionalized Langevin
equation

$\displaystyle\dot{x}(t)=-\left[x(t)-\lambda(t)\right]-\delta_{\mathrm{g}}+\sqrt{2}\
\xi(t)\ ,$ (2)

for scaled effective mass $\delta_{\mathrm{g}}\equiv mg/(\kappa\sigma)$.

Integrating Eq. (2) between measurements from $t$ to $t+t_{\mathrm{s}}$ gives
the discrete dynamics

$\displaystyle
x_{k+1}=x_{k}\operatorname{e}^{-t_{\mathrm{s}}}+\left(1-\operatorname{e}^{-t_{\mathrm{s}}}\right)\left(\lambda_{k}-\delta_{\mathrm{g}}\right)+\sigma_{t_{\mathrm{s}}}\xi_{k}\
,$ (3)

for $x_{k}\equiv x(kt_{\mathrm{s}})$ and
$\lambda_{k}\equiv\lambda(kt_{\mathrm{s}})$. The variance
$\sigma_{t_{\mathrm{s}}}^{2}\equiv 1-\operatorname{e}^{-2t_{\mathrm{s}}}$ of
thermal force fluctuations depends on the sampling interval $t_{\mathrm{s}}$,
and $\xi_{k}$ is a Gaussian random variable with zero mean and covariance
$\expectationvalue{\xi_{k}\xi_{n}}=\delta_{kn}$.

The effect of measurement noise is modeled by a measurement variable $y_{k}$
that is the sum of the bead’s true position and additive white Gaussian noise,

$\displaystyle y_{k}=x_{k}+\sigma_{\mathrm{m}}\nu_{k}\ ,$ (4)

with $\nu_{k}$ a Gaussian random variable with zero mean and covariance
$\expectationvalue{\nu_{k}\nu_{n}}=\delta_{kn}$. We also assume that the
thermal noise affecting the bead’s position is independent of the measurement
noise: $\expectationvalue{\xi_{k}\nu_{n}}=0$, for all $k$ and $n$.

The trap position $\lambda_{k}$ is updated at each time step using a
“ratcheting rule,”

$\displaystyle\lambda_{k+1}=\lambda_{k}+\Theta\left(z_{k}-\lambda_{k}\right)\left[\alpha(z_{k}-\lambda_{k})\right],\
$ (5)

where $\Theta\left(\cdot\right)$ denotes the Heaviside function, $\alpha$ the
feedback gain, and $z_{k}=\\{y_{k},\hat{x}_{k+1}\\}$ the estimate of the
bead’s position (using either the naive measurement $y_{k}$ or the Bayesian
estimate $\hat{x}_{k+1}$). Because of delays, both the naive and Bayesian
estimates of position use information from $y_{k}$ but not $y_{k+1}$; however,
the Bayesian estimate also implicitly incorporates past information
$\\{y_{k-1},y_{k-2},\ldots\\}$ and uses the deterministic component of system
dynamics to predict $y_{k+1}$ (see Eq. (6), below).

Estimating the bead position.—Since the actual bead position $x$ fluctuates on
a scale $\sigma$ and the measurement noise fluctuates on a scale
$\sigma_{\mathrm{m}}$, it is convenient to define a signal-to-noise ratio
$\mathrm{SNR}\equiv\sigma/\sigma_{\mathrm{m}}$ 222SNR is often alternately
defined as a ratio of signal-to-noise power,
$\sigma^{2}/\sigma_{\mathrm{m}}^{2}$.. In a first, _naive_ approach to
designing feedback based on noisy measurements, the feedback rule, Eq. (5),
directly uses the measurement $y_{k}$ to update the trap position
$\lambda_{k+1}$. Notice that this method implicitly estimates the position
$x_{k+1}$ by $y_{k}$. The naive method performs well at high SNR ($\gg 1$) but
poorly at low SNR ($\ll 1$), where a unidirectional ratchet (implemented via
the Heaviside function) often responds to noise rather than actual bead
movements.

In a second, _filtering_ approach, we improve the estimate of $x_{k+1}$ by
using a Fokker-Planck equation to predict the position probability
$p(x_{k+1})$ given $p(x_{k})$, which is itself calculated from measurements up
to $y_{k-1}$. One then updates (or “corrects”) the prediction for time $k+1$
by incorporating the measurement $y_{k}$, using Bayes’ rule (see Appendix).
For systems evolving according to linear dynamics and subject to Gaussian
noise, $p(x_{k})$ remains Gaussian for all $k$ (if $p(x_{0})$ is initialized
as Gaussian, see Appendix) and can be summarized by update equations for the
mean $\hat{x}_{k}$ and variance. The Bayesian filter is then known as the
predictive Kalman filter, and one finds [compare Eq. (3)],

$\displaystyle\hat{x}_{k+1}=\underbrace{\hat{x}_{k}\operatorname{e}^{-t_{\mathrm{s}}}+\left(1-\operatorname{e}^{-t_{\mathrm{s}}}\right)\left(\lambda_{k}-\delta_{\mathrm{g}}\right)}_{\rm
predict}+\underbrace{L\left(y_{k}-\hat{x}_{k}\right)}_{\rm correct}\ ,$ (6)

where the filter gain $L$ “corrects” the naive prediction using the difference
between the previous estimate and actual observation (see Appendix and Ref.
[25, Ch. 8]). The gain $L$ is chosen to minimize the variance
$\expectationvalue{(x_{k}-\hat{x}_{k})^{2}}$ between the true position and its
estimate, and the resulting value is a function of the SNR. This variance is
always less than that of the measurement noise $\sigma_{\mathrm{m}}^{2}$; it
is optimal in that it incorporates all relevant past information contained in
the (long) time series $\\{y_{k},y_{k-1},\ldots\\}$, and no other unbiased
estimator—linear or not—has lower variance [26].

Engine thermodynamics.—Given an estimate of the bead’s position, we infer the
thermodynamic quantities that characterize this engine’s performance. The rate
at which we extract gravitational energy (i.e., change in bead free energy)
during the time interval $[t_{k},t_{k+1})$ is [22]

$\displaystyle\Delta
F_{k+1}=\delta_{\mathrm{g}}\left(\lambda_{k+1}-\lambda_{k}\right)\,,$ (7)

and the time-averaged rate of free-energy change (more informally, the output
power) is $\dot{F}=\tau^{-1}\sum_{k}\Delta F_{k}$, where
$\tau=Nt_{\mathrm{s}}$ is the total duration of an $N$-step protocol. We
estimate the incremental input work (of the trap on the bead) as

$\displaystyle\Delta
W_{k+1}=\frac{1}{2}\left[\left(y_{k+1}-\lambda_{k+1}\right)^{2}-\left(y_{k+1}-\lambda_{k}\right)^{2}\right]\,,$
(8)

which estimates input work based on the noisy measurement $y_{k}$ and not on
the true position $x_{k}$. The Appendix shows that this input-work estimator
is unbiased—as a result of feedback delay.

Results.—A “pure” information engine has zero input trap power:
$\dot{W}~{}=~{}\tau^{-1}\sum_{k}\Delta W_{k}=0$. For error-free measurements
and no feedback delays, this criterion is satisfied at the critical feedback
gain $\alpha^{*}=2$ (see Appendix and Ref. [10]). Physically, this corresponds
to translating the trap to a position opposite its minimum, so that the trap
energy is unchanged. Feedback delay and measurement noise reduce $\alpha^{*}$.
For our experimental conditions with delay of one time step and SNR = 10,
$\dot{W}=0$ is satisfied at the lower feedback gain $\alpha^{*}\approx 1.5$
[Fig. 2(a-b)].

Figure 2: Tuning the feedback gain $\alpha$ to set trap power $\dot{W}=0$. (a)
Trap power for naive (red) and Bayesian (blue) information engines at fixed
SNR = 10. (b) Measured bead ($y(t)$, red) and trap ($\lambda(t)$, black)
trajectories for the naive information engine at SNR = 10. (c) Measured bead
($y(t)$, light red), filtered bead estimate ($\hat{x}(t)$, blue), and trap
($\lambda(t)$, black) trajectories for the Bayesian information engine at SNR
= 2. (b) and (c) have equal scale bars and satisfy $\dot{W}=0$. (d) Critical
feedback gain $\alpha^{*}$ and (e) corresponding input trap power, for naive
(red) and Bayesian (blue) information engines. Hollow red markers denote SNRs
for which $\alpha^{*}>0$ could not be found using the procedure outlined in
(a). Solid red curve in (d) is from numerical simulation (see Appendix).
Experiments here and in other figures have sampling frequency
$\tau_{\mathrm{r}}/t_{\mathrm{s}}=41$, trap stiffness $\kappa=42$ pN/µm,
scaled effective mass $\delta_{\mathrm{g}}=0.8$, diffusion constant $D=0.12$
µm2/s, relaxation time $\tau_{\mathrm{r}}=0.8$ ms, and bead diameter 4 µm.
Markers denote experimental means, and error bars the standard errors of the
mean.

The Bayesian information engine applies feedback based on the filtered
predictive estimate of the bead’s position. As such, it accounts in its
internal model for feedback delays and measurement noise.

Figure 2(a) shows the input trap power, at fixed SNR (=10), as a function of
feedback gain $\alpha$. Despite the delay and finite SNR, the input trap power
is zero for the “ideal” feedback gain $\alpha^{*}=2$. Figure 2(c) illustrates
the operation of the Bayesian filter with a trajectory of the Bayesian
information engine at lower SNR ($=2$). In contrast to the naive information
engine, the trap ratchets only when the estimated position (blue) crosses the
trap center (black), and not necessarily when the noisy measurement (light
red) crosses the trap center.

Next, we investigate how $\alpha^{*}$ depends on SNR, for the naive and
Bayesian information engines. For the naive engine, $\alpha^{*}$ decreases
drastically at low SNR [Fig. 2(d)]. For
$\mathrm{SNR}\lesssim\mathrm{SNR}_{\mathrm{c}}=0.7\pm 0.1$, where
$\mathrm{SNR}_{\mathrm{c}}$ denotes the critical vale of the SNR, a non-zero
$\alpha^{*}$ could not be found experimentally using the procedure outlined in
Fig. 2(a). The Appendix shows that the vanishing of $\alpha^{*}$ corresponds
to a kind of phase transition between a regime where one can set $\dot{F}>0$
while maintaining $\dot{W}=0$ and a regime where one cannot.

By contrast, the critical feedback gain $\alpha^{*}$ remains near 2 for the
Bayesian engine [Fig. 2(d)]. The corresponding measured input trap powers for
both the naive and Bayesian information engines are close to zero relative to
the maximum output power ($\dot{F}_{\mathrm{max}}\approx
0.27\,k_{\mathrm{B}}T/\tau_{\mathrm{r}}$) of the engine, at all SNR [Fig.
2(e)].

Taking advantage of predictions in our estimation algorithm thus simplifies
the experiments, as it eliminates the need to empirically tune the feedback
gain, ensuring that the zero-work condition is always satisfied at $\alpha=2$.
Above, we saw that it also simplifies the work calculations needed to realize
a pure information engine, as the value calculated directly from the noisy
measurement is an unbiased estimator of the true work.

Finally, Fig. 3(a) compares the performance of the naive and Bayesian
information engines, as quantified by the rate of stored gravitational power
$\dot{F}$ while keeping $\dot{W}=0$. Both output powers $\dot{F}$ increase
monotonically with SNR and saturate at the same power level at high SNR
($>10$).

Although the performance of Bayesian and naive information engines is similar
at low and high SNR, there is a striking contrast at intermediate
$\mathrm{SNR}\lesssim 1$. Indeed, the difference of output powers
(Bayesian$-$naive), normalized by $\dot{F}_{\mathrm{max}}$, significantly
exceeds zero for $0.1~{}\leq~{}\mathrm{SNR}~{}\leq 2$ and reaches a maximum at
$\mathrm{SNR}\approx\mathrm{SNR}_{\mathrm{c}}$ (Fig. 3b).

At high SNR, the Bayesian filter “trusts the observation” and returns an
estimate close to the instantaneous measurement, corrected for the expected
bias due to the time delay. Since this bias is small for frequent
measurements, both engines have similar performance and extract all the
favorable thermal fluctuations, saturating at the maximum output power
$\dot{F}_{\mathrm{max}}\approx 0.27$. At low SNR, the measurements are so
noisy that they exceed the scale of the trap. The Bayesian information engine
then extracts negligible power (see Appendix), while the naive engine extracts
zero power. Therefore, at SNR $\gg 1$ and $\ll 1$, the difference of output
powers ($\dot{F}_{\mathrm{B}}-\dot{F}_{\mathrm{N}}$) tends to zero. But at
intermediate SNR, the effective noise averaging in the Bayesian (Kalman)
filter produces more accurate estimates, leading to better feedback decisions
and thus improved engine performance.

Figure 3: Performance of the information engines. a) Output power of naive
(red) and Bayesian (blue) information engines as a function of SNR. Hollow red
markers denote output power at $\alpha^{*}=0$. b) Difference of output work
extraction rates for the Bayesian (B) and naive (N) engines scaled by the
maximum rate ($\dot{F}_{\mathrm{max}}=0.27$). The difference peaks at
SNR=$\mathrm{SNR}_{\mathrm{c}}$ $\approx 0.7$ (vertical dashed lines). Markers
denote experimental means, solid curves the numerical simulations (see
Appendix). Error bars denote a) standard error of the mean and b) propagated
standard error of the mean from a).

To understand why the naive information engine shows a phase transition at a
critical signal-to-noise ratio, we numerically solve a self-consistent
equation for the SNR at which trap power vanishes. Enforcing the condition
that $\alpha^{*}=0$ is the unique solution (see Appendix), we find
$\mathrm{SNR}_{\mathrm{c}}\approx 0.64$, consistent with both numerical
simulations and experiments.

We also find that the phase transition arises from the _biased_ estimate of
the bead’s position from the noisy measurements. This bias has two origins:
the delay due to feedback latency and the failure of the naive measurement to
account for the fact that fluctuations above threshold are rare while noise
fluctuations of either sign are equally likely. Because fluctuations up to the
threshold are rare, the bead is usually below the observed value whenever an
apparent threshold crossing is observed.

By contrast, a phase transition does not occur for the Bayesian information
engine. The Bayesian filter gives an unbiased prediction of the bead’s
position, accounting for both feedback delay and the “prior” associated with
observations near the threshold. As a result, the bead is equally likely to be
on either side of the predicted position, allowing one to tune for zero trap
power and extract at least some power at any SNR value (see Appendix).

Conclusion.—Information engines that decide whether to ratchet using single
noisy measurements have a phase transition at a critical signal-to-noise ratio
$\mathrm{SNR}_{\mathrm{c}}$ and cannot function for
$\mathrm{SNR}<\mathrm{SNR}_{\mathrm{c}}$. By contrast, if its feedback uses a
Bayesian estimate of bead position that incorporates prior measurements, an
information engine can operate at all values of SNR. The maximum performance
benefit over the naive engine occurs at the critical value
$\mathrm{SNR}_{\mathrm{c}}$.

The ability to increase the performance of an information engine at low SNR is
important for experimental investigations of motor mechanisms that use
fluorescent probes [27]. In such applications, lower light intensities for
monitoring fluorescent probes reduce photobleaching and allow longer
measurements of motor behavior.

In addition, using a filtering algorithm to reduce the required accuracy of
information while maintaining a given performance may decrease the
thermodynamic costs of processing position measurements. Generally, a lower
measurement accuracy reduces the minimum thermodynamic costs of running the
controller [28, 14]. However, keeping a memory of past observations should
increase those costs. Further work is needed to understand the efficiency of a
feedback strategy that incorporates a memory of past measurements with one
based purely on the most recent measurement. Such studies could evaluate the
potential performance trade-offs encountered when varying the measurement
accuracy.

###### Acknowledgements.

This research was supported by grant FQXi-IAF19-02 from the Foundational
Questions Institute Fund, a donor-advised fund of the Silicon Valley Community
Foundation. Additional support was from Natural Sciences and Engineering
Research Council of Canada (NSERC) Discovery Grants (D.A.S. and J.B.) and a
Tier-II Canada Research Chair (D.A.S.), an NSERC Undergraduate Summer Research
Award, a BC Graduate Scholarship, and an NSERC Canadian Graduate Scholarship –
Masters (J.N.E.L.).

## Appendix

##  Appendix A Experimental apparatus

The experimental apparatus is designed to optically trap a bead in water. The
optical trap is built by focusing a green laser using a water-immersion
objective (Olympus, 60x 1.2 NA). The position of the trap is controlled using
an acousto-optic deflector (AOD, AA opto-electronics). The position of the
trapped bead is measured by a red detection laser, propagating anti-parallel
to the trapping laser, that is loosely focused in the trapping plane using an
air objective. The intensity of the detection bead is controlled using an
acousto-optic modulator (AOM, AA opto-electronics) to vary the measurement
noise. More details about the experimental apparatus can be found in [10, 29].

Figure S1: Schematic diagram of the experimental apparatus.

##  Appendix B Experimental parameter estimation

The trap stiffness, diffusion constant, and measurement noise are measured by
fitting the power spectrum of the bead’s position to a discrete aliased
Lorentzian [30]. The discrete equation of motion of the bead can be re-written
as (cf. Eq. (3) in the main text)

$\displaystyle x_{k+1}=c\,x_{k}+\Delta x\,\xi_{k}\,,$ (S1)

where $x$ is measured with respect to $\lambda-\delta_{\mathrm{g}}$. The
parameters are

$\displaystyle c\equiv\exp(-t_{\mathrm{s}}/\tau_{\mathrm{r}})=\exp(-2\pi
f_{\mathrm{c}}t_{\mathrm{s}})\,,$ (S2)

and

$\displaystyle\Delta x\equiv\left[\frac{(1-c^{2})D}{2\pi
f_{\mathrm{c}}}\right]^{1/2}\,,$ (S3)

where $D$ is the diffusion constant of the bead, $f_{\mathrm{c}}$ the corner
frequency, and $t_{\mathrm{s}}$ the sampling time. Applying the discrete
Fourier transform to the discrete equation of motion of the bead and
calculating the expected value of the power spectrum $P_{f}$, the discrete
aliased Lorentzian is

$\displaystyle P_{f}^{x}=\frac{2(\Delta x)^{2}t_{\mathrm{s}}}{1+c^{2}-2c\,\cos
2\pi ft_{\mathrm{s}}}\,,$ (S4)

where $f$ indexes the discrete non-negative frequencies. For noisy
measurements

$\displaystyle y=x+\nu\,,$ (S5)

where the measurement noise is Gaussian white noise, the power spectrum of $y$
is

$\displaystyle P_{f}^{y}=P_{f}^{x}+P^{\nu}\,,$ (S6)

where $P^{\nu}$ is the frequency-independent noise floor, and
$\sigma_{m}^{2}=P^{\nu}/2t_{\mathrm{s}}$ the variance of the measurement
noise. Equation (S6) uses the property that the measurement noise is
independent of both the bead’s position $x$ and thermal noise $\xi$.

Figure S2:  Power spectral density $P_{f}^{y}$ of the measured bead position
$y$ in a static trap. Red curve: experiment; blue curve: fit to (S6). The
measured fit parameters are $f_{\mathrm{c}}=167.8\pm 1.6$ Hz, $D=0.114\pm
0.001$ m2/s, and $\sigma_{\mathrm{m}}=0.42\pm 0.06$ nm.

Figure S2 shows the power spectral density of a 4-µm bead in a static trap and
the fit to (S6). The trap stiffness is estimated using $\kappa=2\pi
f_{\mathrm{c}}k_{\mathrm{B}}T/D=40$ pN/µm.

##  Appendix C Performance of Bayesian-filter estimator

When the measurement noise is additive Gaussian white noise and the system
dynamics are linear, as is closely approximated in this experimental setup, an
exactly solvable Bayesian filter can be designed to minimize the least-squares
error between the filter’s state prediction and the true bead position. To
derive the optimal filter for the information engine, we note that the bead’s
motion is a first-order Markov process: its next position $x_{k+1}$ is
conditionally independent of its history given its current position $x_{k}$
and the trap position $\lambda_{k}$,
$p\left(x_{k+1}|x_{k},x_{k-1},\dots,x_{0},\lambda_{k}\right)=p\left(x_{k+1}|x_{k},\lambda_{k}\right)$.
Here, the probability $p\left(x_{k+1}|x_{k},\lambda_{k}\right)$ is obtained
from Eq. (3),

$\displaystyle p$ $\displaystyle\left(x_{k+1}|x_{k},\lambda_{k}\right)\sim$
(S7)
$\displaystyle\mathcal{N}\left(x_{k+1};x_{k}\operatorname{e}^{-t_{\mathrm{s}}}+\left(1-\operatorname{e}^{-t_{\mathrm{s}}}\right)\left(\lambda_{k}-\delta_{\mathrm{g}}\right),\,1-\operatorname{e}^{-2t_{\mathrm{s}}}\right)\
,$

where $\mathcal{N}(x;\mu,\sigma^{2})$ denotes a normal distribution over $x$
with mean $\mu$ and variance $\sigma^{2}$. We also assume that the
measurements of the bead’s position are _memoryless_ : the measurement at the
current time depends only on the current state,
$p\left(y_{k}|x_{k},x_{k-1},\dots,x_{0}\right)=p\left(y_{k}|x_{k}\right)$.
Measurements with additive Gaussian white noise satisfy a distribution of the
form

$\displaystyle
p\left(y_{k}|x_{k}\right)\sim\mathcal{N}\left(y_{k};x_{k},\,\sigma_{\mathrm{m}}^{2}\right)\
.$ (S8)

The standard deviation of the noise distribution is the reciprocal of the
signal-to-noise ratio, $\sigma_{\mathrm{m}}=1/\mathrm{SNR}$, and as such can
be interpreted as a “noise-to-signal” ratio. Using Bayes’ rule, the update
distribution is

$\displaystyle
p\left(x_{k}|Y^{k}\right)=\frac{p\left(y_{k}|x_{k}\right)\,p\left(x_{k}|Y^{k-1}\right)}{p\left(y_{k}|Y^{k-1}\right)}\
,$ (S9)

where $Y^{k}=\\{y_{k},y_{k-1},\dots,y_{1}\\}$ is the set of measurements made
up to and including time $t_{k}$, and $p\left(y_{k}|Y^{k-1}\right)$ is a
normalizing distribution. From this updated distribution and the dynamics
(S7), the predicted distribution at the next discrete time is

$\displaystyle
p\left(x_{k+1}|Y^{k}\right)=\int\differential{x_{k}}\,p\left(x_{k+1}|x_{k}\right)\,p\left(x_{k}|Y^{k}\right)\
.$ (S10)

The measurement update and prediction steps for the information engine can be
formulated as a sequence of transformations of univariate Gaussian probability
densities. Since all densities are Gaussian, we need propagate only their
means and variances:

$\displaystyle\left.\begin{aligned}
\hat{P}_{k}^{y}&=\hat{P}_{k}+\sigma_{\mathrm{m}}^{2}&&\quad\textrm{obs.
var.}\\\
L_{k+1}&=\operatorname{e}^{-t_{\mathrm{s}}}\frac{\hat{P}_{k}}{\hat{P}_{k+1}^{y}}&&\quad\textrm{Kalman
gain}\\\
\hat{P}_{k}^{+}&=-\left(L_{k+1}\right)^{2}\hat{P}_{k+1}^{y}&&\quad\textrm{state
var.}\\\
\hat{x}_{k}^{+}&=L_{k+1}\left(y_{k}-\hat{x}_{k}\right)&&\quad\textrm{state
mean}\\\ \end{aligned}\right\\}\textrm{update}$ (S11a)
$\displaystyle\left.\begin{aligned}
\hat{P}_{k+1}&=\hat{P}_{k}^{+}+\operatorname{e}^{-2t_{\mathrm{s}}}\hat{P}_{k}+\left(1-\operatorname{e}^{-2t_{\mathrm{s}}}\right)&&\quad\textrm{state
var.}\\\
\hat{x}_{k+1}&=\hat{x}_{k}^{+}+\operatorname{e}^{-t_{\mathrm{s}}}\hat{x}_{k}+\left(1-\operatorname{e}^{-t_{\mathrm{s}}}\right)\left(\lambda_{k}-\delta_{\mathrm{g}}\right)&&\quad\textrm{state
mean}\\\ \end{aligned}\right\\}\textrm{predict}$ (S11b)

Intuitively, the filter generates a prediction mean and variance (S11b),
$\hat{x}_{k+1}$ and $\hat{P}_{k+1}$. The stochastic bead dynamics increase the
uncertainty of the bead’s position. This position uncertainty is reduced by
the information obtained from the previous measurement $y_{k}$. The reduction
in prediction uncertainty and corresponding change to the prediction mean is
achieved by feedback based on the discrepancy between the actual measurement
and the prediction, $y_{k}-\hat{x}_{k}$. The Kalman gain $L$ multiplying this
discrepancy accounts for the unreliability of the measurements, given the
scale of measurement noise. The information engine uses this filter’s
prediction mean as the estimate of the bead’s position in the feedback
decision.

##  Appendix D Steady-state Bayesian-filter gain

In experiments, we assume that the experimental parameters do not change with
time. As a result, the means and variances in the time-dependent Eqs. (S11)
quickly approach steady-state values. To obtain the steady-state Bayesian-
filter gain, we first solve the associated Riccati equation [25, Ch. 8] for
the predictive variance,

$\displaystyle\hat{P}^{\mathrm{SS}}$
$\displaystyle=\operatorname{e}^{-2t_{\mathrm{s}}}\hat{P}^{\mathrm{SS}}+\left(1-\operatorname{e}^{-2t_{\mathrm{s}}}\right)-\operatorname{e}^{-2t_{\mathrm{s}}}\frac{\left(\hat{P}^{\mathrm{SS}}\right)^{2}}{\hat{P}^{\mathrm{SS}}+\sigma_{\mathrm{m}}^{2}}\
,$ (S12)

where superscript “SS” without time subscript indicates a steady-state
quantity. The relevant solution to this quadratic equation is

$\displaystyle\hat{P}^{\mathrm{SS}}=\frac{1}{2}$
$\displaystyle\left(1-\sigma_{\mathrm{m}}^{2}\right)\left(1-\operatorname{e}^{-2t_{\mathrm{s}}}\right)$
(S13)
$\displaystyle\times\Bigg{[}1+\sqrt{1+\frac{4\sigma_{\mathrm{m}}^{2}}{\left(1-\sigma_{\mathrm{m}}^{2}\right)^{2}\left(1-\operatorname{e}^{-2t_{\mathrm{s}}}\right)}}\,\,\Bigg{]}\,.$

We discard the other solution, as it is negative and variances must be
positive. The resulting steady-state Bayesian (Kalman) filter gain is

$\displaystyle
L^{\mathrm{SS}}=\frac{\hat{P}^{\mathrm{SS}}}{\hat{P}^{\mathrm{SS}}+\sigma_{m}^{2}}\operatorname{e}^{-t_{\mathrm{s}}}.$
(S14)

Finally, the Kalman-filtered position estimate is

$\displaystyle\hat{x}_{k+1}$
$\displaystyle=L^{\mathrm{SS}}\left(y_{k}-\hat{x}_{k}\right)+\operatorname{e}^{-t_{\mathrm{s}}}\hat{x}_{k}+\left(1-\operatorname{e}^{-t_{\mathrm{s}}}\right)\left(\lambda_{k}-\delta_{\mathrm{g}}\right).$
(S15)

##  Appendix E Experimental implementation of the Bayesian filter

Figure S3:  Experimental performance of the Bayesian-filtered estimate. (a)
Noisy experimental measurement ($y$, red) and the corresponding Bayesian
estimate using Eq. (S15) ($\hat{x}$, blue) of the bead’s position. (b) Power
spectral density and (c) autocorrelation of the noisy measurement (red),
Bayesian estimate (blue), and innovations $\hat{x}-y$ (gray). The experiments
were performed for SNR $\approx$ 2, equilibrium standard deviation
$\sigma=10.1$ nm, and measurement noise $\sigma_{\mathrm{m}}=4.7$ nm.

Experiments on the Bayesian information engine were performed using the
Kalman-filtered position estimate obtained from Eq. (S15). Figure S3 shows the
performance of the experimental implementation of the Kalman filter for a bead
in a static trap. The experiment was performed at SNR $\approx$ 2, where the
experimental parameters were estimated by fitting the power spectral density
using Eq. (S6). Figure S3(a) shows the trajectory of the estimated bead
position (blue), inferred using Eq. (S15) from the noisy measurements (red).

Figure S3(b) shows the power spectral densities of the noisy measurements
(red) and Bayesian estimates (blue). The power spectral density of the noisy
measurements saturates at a finite noise floor (PSD$~{}=~{}10^{-3}$ nm2/Hz).
The filter-estimated position (blue) has a noise floor less then a tenth that
of the individual noisy measurements (red). The power spectral density of the
innovation $\hat{x}_{n}-y_{n}$ is flat, showing that the innovations follow a
white-noise distribution, and hence all the position correlations arising from
the bead’s relaxation in the harmonic trap are extracted by the estimator.

Figure S3(c) shows the normalized autocorrelation of the noisy measurements
(red), Bayesian estimates (blue), and innovations (gray). From Eq. (S5), the
autocorrelation of the noisy measurements is

$\displaystyle\expectationvalue{y_{n}y_{m}}$
$\displaystyle=\expectationvalue{(x_{n}+\nu_{n})(x_{m}+\nu_{m})}$ (S16a)
$\displaystyle=\sigma_{\mathrm{m}}^{2}\delta_{nm}+\operatorname{e}^{(m-n)t_{\mathrm{s}}},$
(S16b)

where we have again used the property that the measurement noise is
independent of the true bead position.

The noisy measurements (red curve) show a delta-function peak at $t=0$ on top
of a broader exponential decay, reflecting two processes: First, there is the
contribution of measurement noise, which is delta correlated; second, there is
the contribution of the dynamics, which has an exponential decay of
correlations.

By contrast, the Bayesian estimate (blue curve) has only an exponential decay
due to the dynamics—as would the autocorrelation function of noise-free
dynamics. In other words, the Bayesian estimator distills the dynamical
information in the correlated signal. The innovations (gray curve) are almost
entirely delta correlated. Indeed, the degree to which the innovations
approximate a delta function serves as a measure of the quality of the
Bayesian filter. For example, deviations between a parameter value in the
physical system and the corresponding model used for the Bayesian filter
produce dynamical correlations in the innovations autocorrelation function
[25]. Indeed, this mismatch has been used as a way to empirically tune
parameters for Bayesian (Kalman) filters (“innovation whitening”) [31].

##  Appendix F Numerical-simulation methods

The numerical simulations generate multiple long trajectories of the bead
using the discrete-time dynamics,

$\displaystyle
x_{k+1}=x_{k}\operatorname{e}^{-t_{\mathrm{s}}}+\left(1-\operatorname{e}^{-t_{\mathrm{s}}}\right)\left(\lambda_{k}-\delta_{\mathrm{g}}\right)+\sigma_{t_{\mathrm{s}}}\xi_{k}\
,$ (S17)

where $x_{k}\equiv x(kt_{\mathrm{s}})$ and
$\lambda_{k}\equiv\lambda(kt_{\mathrm{s}})$. The corresponding noisy
measurement is $y_{k}=x_{k}+\sigma_{\mathrm{m}}\nu_{k}$, where $\nu_{k}$ is a
Gaussian random variable with zero mean and covariance
$\expectationvalue{\nu_{k}\nu_{n}}=\delta_{kn}$. For the naive information
engine, the trap position is updated using the feedback rule

$\displaystyle\lambda_{k+1}=\lambda_{k}+\Theta\left(y_{k}-\lambda_{k}\right)\,\alpha(y_{k}-\lambda_{k}).$
(S18)

For the Bayesian information engine, the trap position is updated using the
feedback rule

$\displaystyle\lambda_{k+1}=\lambda_{k}+\Theta\left(\hat{x}_{k+1}-\lambda_{k}\right)\,\alpha(\hat{x}_{k+1}-\lambda_{k})\
,$ (S19)

using the Bayesian estimate (S15) of the bead’s position. For each engine
type, the free-energy change is estimated using Eq. (7) and the work using the
unbiased estimator Eq. (8) in the main text.

##  Appendix G Trap-work estimates are unbiased

Here, we show that our empirical estimator for the input trap work

$\displaystyle
W_{k+1}=\frac{1}{2}\left[\left(y_{k+1}-\lambda_{k+1}\right)^{2}-\left(y_{k+1}-\lambda_{k}\right)^{2}\right]$
(S20)

is an unbiased estimator when the feedback rule follows

$\displaystyle\lambda_{k+1}=\lambda_{k}+\Theta\left(z_{k}-\lambda_{k}\right)\,\alpha(z_{k}-\lambda_{k})\
.$ (S21)

Note that this feedback rule updates the value of the control parameter
$\lambda_{k}\to\lambda_{k+1}$ at time $t_{k+1}$ according to the estimate
$z_{k}$ of the bead’s position at the _previous_ time $t_{k}$, not the
measurement that is made at the current time $t_{k+1}$. In other words, there
is a delay of one cycle between when a measurement is taken and when the
corresponding feedback is applied.

For convenience, we introduce a new notation where the superscript on the
thermodynamic quantities (e.g., $W_{k}^{x}$) denotes the type of measurement
used in the estimation of the thermodynamic quantity. In this notation, we
write (S20) as

$\displaystyle
W_{k+1}^{y}=\frac{1}{2}\left[\left(y_{k+1}-\lambda_{k+1}\right)^{2}-\left(y_{k+1}-\lambda_{k}\right)^{2}\right]\
,$ (S22)

and the true estimate of the work from stochastic thermodynamics [32] given by
the actual potential-energy change when updating $\lambda_{k}$ to
$\lambda_{k+1}$,

$\displaystyle
W_{k+1}^{x}=\frac{1}{2}\left[\left(x_{k+1}-\lambda_{k+1}\right)^{2}-\left(x_{k+1}-\lambda_{k}\right)^{2}\right]\
.$ (S23)

Figure S4: Numerical simulations of bias in the trap-power estimates from
noisy measurements, as a function of SNR. a) Estimated trap power in naive
information engine with no feedback delay (latency) and applying feedback
based on the most recent noisy measurement:
$\lambda_{k+1}=\lambda_{k}+\Theta\left(y_{k+1}-\lambda_{k}\right)\left[\alpha(y_{k+1}-\lambda_{k})\right]$.
b), c) Estimated trap power when feedback is applied after a delay of one time
step, using the feedback rule (5) in the main text, based on (b) biased naive
$z_{k}=y_{k}$ or (c) unbiased filtered $z_{k}=\hat{x}_{k+1}$ position
estimates. Solid curve: estimate using true positions $x$. Markers: estimate
using measurements $y_{k}$. The numerical simulations were performed for
sampling frequency $f_{\mathrm{s}}=40$, feedback gain $\alpha=2$, and scaled
mass $\delta_{\mathrm{g}}=0.84$.

Now we compare the empirical estimates to the true values of the thermodynamic
quantities. Expanding the definition of the empirical input work (S20) using
$y_{k}=x_{k}+\nu_{k}$, where $\nu_{k}$ is a zero-mean Gaussian random variable
with variance $\sigma^{2}_{\mathrm{m}}$, gives

$\displaystyle W_{k+1}^{y}$
$\displaystyle=\frac{1}{2}\left[\left(y_{k+1}-\lambda_{k+1}\right)^{2}-\left(y_{k+1}-\lambda_{k}\right)^{2}\right]$
(S24a)
$\displaystyle=\frac{1}{2}\left[\left(x_{k+1}-\lambda_{k+1}+\nu_{k+1}\right)^{2}\right.$
(S24b) $\displaystyle\hskip
85.35826pt\left.-\left(x_{k+1}-\lambda_{k}+\nu_{k+1}\right)^{2}\right]$
$\displaystyle=\frac{1}{2}\left[\left(x_{k+1}-\lambda_{k+1}\right)^{2}-\left(x_{k+1}-\lambda_{k}\right)^{2}\right]$
(S24c) $\displaystyle\hskip
113.81102pt+\left(\lambda_{k+1}-\lambda_{k}\right)\nu_{k+1}$
$\displaystyle=\left(W_{k+1}\right)^{x}+\left(\lambda_{k}-\lambda_{k+1}\right)\nu_{k+1}\
.$ (S24d)

Averaging over the noise at time $t_{k+1}$, when the trap position has a
nonnegative update $\lambda_{k+1}-\lambda_{k}>0$ (when
$\lambda_{k+1}=\lambda_{k}$, the work is trivially $W=0$ regardless of which
position estimate is used), we find that

$\displaystyle\expectationvalue{W_{k+1}^{y}}_{\nu}$
$\displaystyle=\expectationvalue{W_{k+1}^{x}}_{\nu}+\expectationvalue{\left(\lambda_{k}-\lambda_{k+1}\right)\nu_{k+1}}$
(S25a)
$\displaystyle=\expectationvalue{W_{k+1}^{x}}_{\nu}+\expectationvalue{\left[\lambda_{k}-(\lambda_{k}+\alpha(z_{k}-\lambda_{k}))\right]\nu_{k+1}}$
(S25b)
$\displaystyle=\expectationvalue{W_{k+1}^{x}}_{\nu}-\alpha\expectationvalue{z_{k}\
\nu_{k+1}}+\alpha\expectationvalue{\lambda_{k}\ \nu_{k+1}}$ (S25c)
$\displaystyle=\expectationvalue{W_{k+1}^{x}}_{\nu}\ ,$ (S25d)

where, in the last step, we have used the property that both the position
estimate $z_{k}$ (which denotes the noisy measurement $y_{k}$ in the naive
case and the predictive estimate $\hat{x}_{k+1}$ in the Bayesian case), and
the trap position $\lambda_{k}$ at time step $k$ are independent of the
_future_ measurement noise $\nu_{k+1}$. Without the delay, the variables are
correlated and the measurements are biased (Fig. S4a). This calculation
suggests that feedback delay actually eliminates the bias in the empirical
estimate of the trap work. Using numerical simulations, we compute the
empirical power $\dot{W}^{y}$ and find that it agrees well with the true
values $\dot{W}^{x}$ for both the naive (Fig. S4b) and Bayesian (Fig. S4c)
information engines.

We also note that the trap work could alternatively be estimated using the
Bayesian filtered position; however, the estimator in Eq. (S20) is robustly
unbiased: It is unbiased even if the dynamical model used in the Kalman filter
uses parameters that differ from those describing the physical system. The
calculation of the trap position $\lambda_{k}$ is based on past information
and—whatever the algorithm used to compute its value—will always be
independent of the current measurement.

##  Appendix H Experiment and simulation of the critical feedback gain
$\alpha^{*}$

Figure S5: Experiments (markers) and simulations (curves) from Fig. 2 in the
main text, presented on a logarithmic ordinate. (a) The critical feedback gain
$\alpha^{*}$, and the corresponding (b) output power $\dot{F}$ and (c) trap
power $\dot{W}$ of the naive (red) and Bayesian (blue) information engines,
each as a function of SNR. Hollow markers denote experiments at
$\mathrm{SNR}<\mathrm{SNR}_{\mathrm{c}}$. Error bars denote the standard error
of the mean.

Figure S5 presents Fig. 2 in the main text, but with a logarithmic ordinate.
According to simulations, for the naive information engine the critical
feedback gain $\alpha^{*}=0$ is the only solution below a critical signal-to-
noise ratio $\mathrm{SNR}_{\mathrm{c}}$, as shown in Fig. S5(a). At low SNR,
experimental determinations of the critical feedback gain $\alpha^{*}$ do not
agree with numerical simulations (Fig. S5(a)), denoted as red hollow markers.
Although experiments find $\alpha^{*}>0$, the naive output powers $\dot{F}$
are still an order of magnitude smaller than those of the Bayesian information
engine (Fig. S5(b)). Figure S5(c) shows the corresponding input trap powers
$\dot{W}$, which remain small at low SNR.

##  Appendix I Theoretical arguments for the phase transition in the naive
information engine

Empirically, the naive estimator can extract free energy only above a critical
signal-to-noise ratio $\mathrm{SNR}_{\mathrm{c}}~{}\approx~{}0.7$.

We define the critical SNR as the highest value for which $\alpha^{*}=0$ is
the only solution for which $\dot{W}=0$. Note that $\alpha=0$ corresponds to
never moving the trap and therefore not doing any work on the bead. When no
feedback is performed, the stationary distribution of the bead inside the trap
is the equilibrium Boltzmann distribution,

$\displaystyle
p(x_{k}|\lambda_{k})=\mathcal{N}(x_{k};\lambda_{k}-\delta_{\mathrm{g}},\,1)\,.$
(S26)

This equation, the bead’s propagator $p_{x}(x_{k+1}|x_{k},\lambda_{k})$ in Eq.
(S7), and the measurement distribution $p_{y}(y_{k}|x_{k})$ in Eq. (S8)
together allow us to evaluate the input work,

$\displaystyle\left\langle\Delta W_{k+1}\right\rangle$
$\displaystyle=\frac{1}{2}\left\langle\left(x_{k+1}-\lambda_{k+1}\right)^{2}-\left(x_{k+1}-\lambda_{k}\right)^{2}\right\rangle$
(S27a)
$\displaystyle=\frac{1}{2}\left\langle\left[x_{k+1}-\lambda_{k}-\Theta(y_{k}-\lambda_{k})\,\alpha(y_{k}-\lambda_{k})\right]^{2}-\left(x_{k+1}-\lambda_{k}\right)^{2}\right\rangle$
(S27b)
$\displaystyle=\frac{1}{2}\int\differential{\lambda_{k}}p(\lambda_{k})\int\differential{x_{k+1}}\int\differential{x_{k}}\int\limits_{\lambda_{k}}^{\infty}\differential{y}\,p_{x}(x_{k+1}|x_{k},\lambda_{k})\,p_{y}(y_{k}|x_{k})\,p(x_{k}|\lambda_{k})$
$\displaystyle\hskip
170.71652pt\times\left\\{\left[x_{k+1}\\!-\\!\lambda_{k}-\alpha\left(y_{k}\\!-\\!\lambda_{k}\right)\right]^{2}-\left(x_{k+1}\\!-\\!\lambda_{k}\right)^{2}\right\\}$
(S27c)
$\displaystyle=\frac{1}{2}\int\mathrm{d}\lambda_{k}\,p(\lambda_{k})\int\mathrm{d}\tilde{x}^{+}\int\mathrm{d}\tilde{x}\int\limits_{0}^{\infty}\mathrm{d}\tilde{y}\,p_{x}(\tilde{x}^{+}\\!+\\!\lambda_{k}|\tilde{x}+\lambda_{k},\lambda_{k})\,p_{y}(\tilde{y}+\lambda_{k}|\tilde{x}+\lambda_{k})\,p(\tilde{x}+\lambda_{k}|\lambda_{k})$
$\displaystyle\hskip
284.52756pt\times\left[\left(\tilde{x}^{+}-\alpha\tilde{y}\right)^{2}-\left(\tilde{x}^{+}\right)^{2}\right]$
(S27d)
$\displaystyle=\frac{1}{2}\int\mathrm{d}\lambda_{k}\,p(\lambda_{k})\int\mathrm{d}\tilde{x}^{+}\int\mathrm{d}\tilde{x}\int\limits_{0}^{\infty}\mathrm{d}\tilde{y}\,\mathcal{N}\left[\tilde{x}^{+};\tilde{x}\operatorname{e}^{-t_{\mathrm{s}}}-\delta_{\mathrm{g}}(1-\operatorname{e}^{-t_{\mathrm{s}}}),\,1-\operatorname{e}^{-2t_{\mathrm{s}}}\right]\,\mathcal{N}(\tilde{y};\tilde{x},\,\sigma_{\mathrm{m}}^{2})\,\mathcal{N}(\tilde{x};-\delta_{\mathrm{g}},1)$
$\displaystyle\hskip
284.52756pt\times\left[\left(\tilde{x}^{+}-\alpha\tilde{y}\right)^{2}-\left(\tilde{x}^{+}\right)^{2}\right]$
(S27e)
$\displaystyle=\frac{\alpha}{4}\left[\alpha\left(\delta_{\mathrm{g}}^{2}+1+\sigma_{\mathrm{m}}^{2}\right)-2\delta_{\mathrm{g}}^{2}-2\operatorname{e}^{-t_{\mathrm{s}}}\right]\left[1-\erf\left(\frac{\delta_{\mathrm{g}}}{\sqrt{2(1+\sigma_{\mathrm{m}}^{2})}}\right)\right]+\sqrt{\frac{1+\sigma_{\mathrm{m}}^{2}}{8\pi}}\,\alpha(2-\alpha)\delta_{\mathrm{g}}\exp\left[-\frac{\delta_{\mathrm{g}}^{2}}{2(1+\sigma_{\mathrm{m}}^{2})}\right]\,,$
(S27f)

where we substituted $\tilde{x}=x_{k}-\lambda_{k}$,
$\tilde{y}=y_{k}-\lambda_{k}$, and $\tilde{x}^{+}=x_{k+1}-\lambda_{k}$ in line
(S27d), and used the normalization of the unknown $p(\lambda_{k})$ in line
(S27f).

Solving $\left\langle\Delta W_{k+1}\right\rangle=0$ for $\alpha$ yields
solutions $\alpha^{*}=0$, and

$\displaystyle\alpha^{*}$
$\displaystyle=2\frac{\sqrt{2(1+\sigma_{\mathrm{m}}^{2})}\,\delta_{\mathrm{g}}\exp\left[-\frac{\delta_{\mathrm{g}}^{2}}{2(1+\sigma_{\mathrm{m}}^{2})}\right]-\sqrt{\pi}\left(\delta_{\mathrm{g}}^{2}+\operatorname{e}^{-t_{\mathrm{s}}}\right)\left[1-\erf\frac{\delta_{\mathrm{g}}}{\sqrt{2(1+\sigma_{\mathrm{m}}^{2})}}\right]}{\sqrt{2(1+\sigma_{\mathrm{m}}^{2})}\,\delta_{\mathrm{g}}\exp\left[-\frac{\delta_{\mathrm{g}}^{2}}{2(1+\sigma_{\mathrm{m}}^{2})}\right]-\sqrt{\pi}\left(\delta_{\mathrm{g}}^{2}+1+\sigma_{\mathrm{m}}^{2}\right)\left[1-\erf\frac{\delta_{\mathrm{g}}}{\sqrt{2(1+\sigma_{\mathrm{m}}^{2})}}\right]}\,.$
(S28)

At the critical $\sigma_{\mathrm{m}}^{2}$, the solution $\alpha^{*}$
bifurcates. We determine the critical SNR from Eq. (S28) by solving

$\displaystyle\alpha^{*}=0$ (S29)

for $\sigma_{\mathrm{m}}^{2}$, which can only be done numerically. For
$t_{\mathrm{s}}=1/41$ and $\delta_{\mathrm{g}}=0.8$, the parameters used in
Figs. 2 and 3,

$\displaystyle\mathrm{SNR}_{\mathrm{c}}=\left(\sigma_{\mathrm{m}}^{\mathrm{c}}\right)^{-1}\approx
0.64\,.$ (S30)

Figure 2 in the main text shows that this critical SNR indeed empirically
coincides with the emergence of a positive $\alpha^{*}$. We note that the
accuracy of this theoretical estimate (and thus the numerical precision we
give) is limited by the accuracy of experimental parameters such as
$\delta_{g}$ and $t_{\mathrm{s}}$ which depend on the bead size.

##  Appendix J Experimental evidence supporting a phase transition in the
naive information engine

In this section, we present detailed experimental evidence for a phase
transition in the naive information engine. Figure S6 (a) shows the trap power
as a function of the feedback gain $\alpha$ for different SNR, where the
markers in the blue shaded region represent SNR values for which only positive
trap powers ($\dot{W}>0$) were measured. The experimental points are fit to a
quadratic function; Fig. S6 (b) shows the linear coefficients of the fits in
Fig. S6 (a) as a function of SNR. The measured slope switches from negative at
high $\mathrm{SNR}\gtrsim 0.8$ to positive at $\mathrm{SNR}\lesssim 0.6$,
consistent with $\mathrm{SNR}_{\mathrm{c}}$ determined in Eq. (S30). To
measure the critical signal-to-noise ratio (where the linear coefficient is
zero), the linear coefficients for SNRs 0.4–0.8 are linearly fit, giving an
estimated zero linear coefficient and hence $\mathrm{SNR}_{\mathrm{c}}$ at
$0.7\pm 0.1$. In the naive information engine experiments, for
$\mathrm{SNR}<\mathrm{SNR}_{\mathrm{c}}$, we set $\alpha>\alpha^{*}$ because
we cannot find an $\alpha$ that leads to $\dot{W}<0$. Thus, the $\alpha^{*}$
we report in such cases (red hollow markers in Fig. 3, main text) is an upper
bound on the $\alpha^{*}$ that satisfies $\dot{W}\gtrsim 0$: Either there is a
smaller positive $\alpha^{*}$ that we could not detect experimentally because
differences in $\dot{W}$ are too close to zero, or the only value that
(always) enforces $\dot{W}=0$ is $\alpha^{*}=0$. The corresponding $\dot{F}$
measured using the upper-bound value of $\alpha^{*}$ then is an upper bound on
the output power for $\mathrm{SNR}<\mathrm{SNR}_{\mathrm{c}}$.

Figure S6: Phase transition in naive information engine. (a) Trap power as a
function of feedback gain $\alpha$ for different signal-to-noise ratios: 0.4,
0.5, 0.6, 0.8, 1.2, 1.3, 2, and 6 (different shades of gray, scale bar in
(b)). Markers: experiment; curves: fit to quadratic functions. Blue shaded
region: SNR values for which $\dot{W}>0$ for all values of $\alpha$. (b)
Linear coefficients from quadratic fits (curves in (a)), for different SNRs.
Blue line: linear fit to first four points. Error bars each denote standard
error of the mean in (a) and parameter fit error in (b).

Importantly, the experimental trap powers in Fig. S6 need not match the work
in Eq. (S27f) because that equation assumed that the bead’s stationary-state
position distribution is the equilibrium distribution (S26), which only
strictly holds when $\alpha=0$.

##  Appendix K The Bayesian information engine has no phase transition

Here, we present theoretical arguments to support the claim that there is no
corresponding “phase transition” for the information engine that estimates the
bead position using filtered measurements.

###  K.1 Feedback gain

First, we show that feedback gain $\alpha=2$ generally produces vanishing
input work. Using the predictive Kalman filter [Eq. (6)] in the feedback rule
[Eq. (5)] with $\alpha=2$, the trap work is

$\displaystyle\left\langle\Delta W_{k+1}\right\rangle$
$\displaystyle=\frac{1}{2}\left\langle\left(x_{k+1}-\lambda_{k+1}\right)^{2}-\left(x_{k+1}-\lambda_{k}\right)^{2}\right\rangle$
(S31a)
$\displaystyle=\frac{1}{2}\left\langle\left[x_{k+1}-\lambda_{k}-2\,\Theta(\hat{x}_{k+1}-\lambda_{k})\,(\hat{x}_{k+1}-\lambda_{k})\right]^{2}-\left(x_{k+1}-\lambda_{k}\right)^{2}\right\rangle$
(S31b)
$\displaystyle=\frac{1}{2}\int\mathrm{d}\lambda_{k}\,p(\lambda_{k})\int\mathrm{d}x_{k+1}\int\limits_{\lambda_{k}}^{\infty}\mathrm{d}\hat{x}_{k+1}\,p(x_{k+1},\hat{x}_{k+1}|\lambda_{k})\left\\{\left[x_{k+1}\\!-\\!\lambda_{k}-2\left(\hat{x}_{k+1}\\!-\\!\lambda_{k}\right)\right]^{2}-\left(x_{k+1}\\!-\\!\lambda_{k}\right)^{2}\right\\}$
(S31c)
$\displaystyle=2\int\mathrm{d}\lambda_{k}\,p(\lambda_{k})\int\limits_{\lambda_{k}}^{\infty}\mathrm{d}\hat{x}_{k+1}\,p(\hat{x}_{k+1}|\lambda_{k})\,\left(\hat{x}_{k+1}-\lambda_{k}\right)\underbrace{\int\differential{x_{k+1}}\,p(x_{k+1}|\hat{x}_{k+1},\lambda_{k})\,\left(x_{k+1}-\hat{x}_{k+1}\right)}_{=\,0}$
(S31d) $\displaystyle=0\,,$ (S31e)

where (S31d) used the fact that the posterior distribution
$p(x_{k+1}|\hat{x}_{k+1},\lambda_{k})$ of the true bead position $x_{k+1}$ is
symmetric around the filter estimate $\hat{x}_{k+1}$, which in this case is
the mean of the Gaussian posterior. Figure S7 shows the input work as a
function of the feedback gain $\alpha$ for different signal-to-noise ratios.
$\alpha=2$ makes the input work vanish.

###  K.2 Free-energy gain

With feedback gain $\alpha=2$ and the feedback rule in Eq. (5), the free
energy-gain per time step, Eq. (7), becomes

$\displaystyle\left\langle\Delta
F_{k+1}\right\rangle=\delta_{\mathrm{g}}\left\langle\lambda_{k+1}-\lambda_{k}\right\rangle$
(S32a)
$\displaystyle\quad=2\delta_{\mathrm{g}}\int\mathrm{d}\lambda_{k}\,p(\lambda_{k})\underbrace{\int\limits_{\lambda_{k}}^{\infty}\mathrm{d}\hat{x}_{k+1}\,p(\hat{x}_{k+1}|\lambda_{k})\,\left(\hat{x}_{k+1}\\!-\\!\lambda_{k}\right)}_{>\,0}$
(S32b) $\displaystyle\quad>0\,,$ (S32c)

where we used the fact that the probability density
$p(\hat{x}_{k+1}|\lambda_{k})$ is positive and always has a tail (however
small) reaching into the region $\hat{x}_{k+1}>\lambda_{k}$. Hence, there is
no phase transition in the Bayesian information engine. Note that for the
naive engine

$\displaystyle\langle\Delta
F_{k+1}\rangle~{}=~{}\alpha\delta_{\mathrm{g}}\int\mathrm{d}\lambda_{k}\,p(\lambda_{k})\int_{\lambda_{k}}^{\infty}\mathrm{d}y_{k}\,p(y_{k}|\lambda_{k})(y_{k}-\lambda_{k})\,,$
(S33)

which vanishes for $\alpha=0$.

Figure S7: Experiments showing the Bayesian information engine has no phase
transition. Trap power $\dot{W}$ as a function of feedback gain $\alpha$ for
different signal-to-noise ratios (different shades of gray). Solid curves
connect solid markers to ease visualization. Error bars each denote the
standard error of the mean.

To corroborate the finding that the Bayesian information engine has no phase
transition, we calculate the low-SNR limit of the free-energy gain per time
step. When SNR is low, feedback is applied rarely and hence the stationary
distribution of bead positions approximates the equilibrium Boltzmann
distribution,
$p(x_{k+1}|\lambda_{k})=\mathcal{N}(x_{k+1};\lambda_{k}-\delta_{\mathrm{g}},1)$.
Hence, the distribution $\hat{p}(\hat{x}_{k+1}|\lambda_{k})$ of filter
estimates is also approximately Gaussian,

$\displaystyle\hat{p}(\hat{x}_{k+1}|\lambda_{k})=\mathcal{N}\left(\hat{x}_{k+1};\hat{\mu},\,\hat{c}\right)\,.$
(S34)

The mean and variance (S11b) of the predictive Kalman filter dictate the
steady-state the posterior distribution, of the true bead position $x_{k+1}$
given the filter estimate $\hat{x}_{k+1}$ ,

$\displaystyle
p(x_{k+1}|\hat{x}_{k+1})=\mathcal{N}\left(x_{k+1};\hat{x}_{k+1},\,\hat{P}^{\mathrm{SS}}\right)\,,$
(S35)

allowing us to solve for $\hat{p}(\hat{x}_{k+1}|\lambda_{k})$,

$\displaystyle\hat{p}(\hat{x}_{k+1}|\lambda_{k})=\int\differential{u}\,p(x_{k+1}|\hat{x}_{k+1}=u)\,\hat{p}(u|\lambda_{k})\,,$
(S36)

such that

$\displaystyle\hat{\mu}$ $\displaystyle=\lambda_{k}-\delta_{\mathrm{g}}$
(S37a) $\displaystyle\hat{c}$ $\displaystyle=1-\hat{P}^{\mathrm{ss}}\,,$
(S37b)

are the mean and variance of the filter state in Eq. (S36).

Inserting these cumulants into Eq. (S34) and evaluating Eq. (S32b) again using
the normalization of $p(\lambda_{k})$ gives

$\displaystyle\left\langle\Delta F_{k+1}\right\rangle$
$\displaystyle=\delta_{\mathrm{g}}\sqrt{\frac{2(1-\hat{P}^{\mathrm{ss}})}{\pi}}\,\exp\left[-\frac{\delta_{\mathrm{g}}^{2}}{2(1-\hat{P}^{\mathrm{ss}})}\right]-\delta_{\mathrm{g}}^{2}\left[1-\erf\frac{\delta_{\mathrm{g}}}{\sqrt{2(1-\hat{P}^{\mathrm{ss}})}}\right]\,.$
(S38)

Expanding the stationary filter variance (S13) for small
$\mathrm{SNR}=1/\sigma_{\mathrm{m}}$ yields

$\displaystyle\hat{P}^{\mathrm{ss}}$
$\displaystyle=1-\frac{\mathrm{SNR}^{2}}{\operatorname{e}^{2t_{\mathrm{s}}}-1}+\mathcal{O}(\mathrm{SNR}^{4})$
(S39)

Then, with $\erf(x)\approx
1-\operatorname{e}^{-x^{2}}\left(1/x-1/2x^{3}\right)/\sqrt{\pi}$ for
$x\to\infty$,

$\displaystyle\left\langle\Delta F_{k+1}\right\rangle$
$\displaystyle\approx\sqrt{\frac{2}{\pi}}\left(\frac{\mathrm{SNR}}{\sqrt{\operatorname{e}^{2t_{\mathrm{s}}}-1}}\right)^{3}\exp\left[-\frac{\delta_{\mathrm{g}}^{2}(\operatorname{e}^{2t_{\mathrm{s}}}-1)}{2\,\mathrm{SNR}^{2}}\right]>0\,,$
(S40)

illustrating that the Bayesian information ratchet extracts free energy at all
signal-to-noise ratios.

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$\displaystyle=\mathbb{E}\left[\overline{\Sigma}^{1/2}\cdot\frac{\overline{\Sigma}^{3/2}}{\gamma^{*}-\mathbb{E}\left[\frac{{\mathcal{T}}(\overline{Y})}{a^{*}-{\mathcal{T}}(\overline{Y})}\right]\overline{\Sigma}}\right]^{2}$
$\displaystyle\leq\mathbb{E}\left[\overline{\Sigma}\right]\mathbb{E}\left[\frac{\overline{\Sigma}^{3}}{\left(\gamma^{*}-\mathbb{E}\left[\frac{{\mathcal{T}}(\overline{Y})}{a^{*}-{\mathcal{T}}(\overline{Y})}\right]\overline{\Sigma}\right)^{2}}\right].$
(G.1)

Rearranging terms and noting that the common factor
$\frac{1}{\delta}\mathbb{E}\left[\left(\frac{{\mathcal{T}}(\overline{Y})}{a^{*}-{\mathcal{T}}(\overline{Y})}\right)^{2}\right]$
in the last two terms is positive, we conclude the proof of the proposition. ∎

######  Proposition G.2 (Stein’s lemma [Ste74, Ste81]).

Let $W\sim{\mathcal{N}}(0,\sigma^{2})$ and let
$f\colon{\mathbb{R}}\to{\mathbb{R}}$ be such that both expectations below
exist. Then

$\displaystyle\mathbb{E}\left[Wf(W)\right]$
$\displaystyle=\sigma^{2}\mathbb{E}\left[f^{\prime}(W)\right].$

######  Proposition G.3.

Let $W\sim{\mathcal{N}}(0_{d},\sigma^{2}I_{d})$. Let
$B\in{\mathbb{R}}^{d\times d}$ denote a sequence of deterministic matrices
such that the empirical spectral distribution of $\frac{1}{d}B$ converges to
the law of a random variable $\overline{\Sigma}$. Then

$\displaystyle\lim_{d\to\infty}\frac{1}{d}\mathbb{E}\left[W^{\top}BW\right]$
$\displaystyle=\sigma^{2}\mathbb{E}\left[\overline{\Sigma}\right].$

###### Proof.

The proof follows from a straightforward calculation:

$\displaystyle\lim_{d\to\infty}\frac{1}{d}\mathbb{E}\left[W^{\top}BW\right]$
$\displaystyle=\lim_{d\to\infty}\frac{1}{d}\sum_{(i,j)\in[d]^{2}}\mathbb{E}\left[B_{i,j}W_{i}W_{j}\right]$
$\displaystyle=\lim_{d\to\infty}\frac{1}{d}\sum_{i\in[d]}\mathbb{E}\left[W_{i}^{2}\right]B_{i,i}=\lim_{d\to\infty}\frac{\sigma^{2}}{d}\operatorname{Tr}(B)=\sigma^{2}\mathbb{E}\left[\overline{\Sigma}\right].\qed$
(G.2)

######  Proposition G.4.

Let

$\displaystyle(G,H)$
$\displaystyle\sim{\mathcal{N}}\left(\begin{bmatrix}0_{d}\\\
0_{d}\end{bmatrix},\begin{bmatrix}\sigma^{2}&\rho\\\
\rho&\tau^{2}\end{bmatrix}\otimes I_{d}\right).$

Let $B\in{\mathbb{R}}^{d\times d}$ denote a sequence of deterministic matrices
such that the empirical spectral distribution of $\frac{1}{d}B$ converges to
the law of a random variable $\overline{\Sigma}$. Then

$\displaystyle\lim_{d\to\infty}\frac{1}{d}\mathbb{E}\left[G^{\top}BH\right]$
$\displaystyle=\rho\mathbb{E}\left[\overline{\Sigma}\right].$

###### Proof.

The proof follows from a straightforward calculation:

$\displaystyle\lim_{d\to\infty}\frac{1}{d}\mathbb{E}\left[G^{\top}BH\right]$
$\displaystyle=\lim_{d\to\infty}\frac{1}{d}\sum_{(i,j)\in[d]^{2}}\mathbb{E}\left[B_{i,j}G_{i}H_{j}\right]$
$\displaystyle=\lim_{d\to\infty}\frac{1}{d}\sum_{i\in[d]}\mathbb{E}\left[G_{i}H_{i}\right]B_{i,j}=\lim_{d\to\infty}\frac{\rho}{d}\operatorname{Tr}(B)=\rho\mathbb{E}\left[\overline{\Sigma}\right].\qed$
(G.3)

######  Proposition G.5 (Implicit function theorem).

Let $H\colon{\mathbb{R}}^{2}\to{\mathbb{R}}$ be a continuously differentiable
function. Fix a point $(x_{0},y_{0})\in{\mathbb{R}}^{2}$ with
$H(x_{0},y_{0})=0$. If $\frac{\partial}{\partial y}H(x_{0},y_{0})\neq 0$, then
there exists an open interval ${\mathcal{U}}\subset{\mathbb{R}}$ containing
$x_{0}$ such that there exists a unique continuously differentiable function
$h\colon{\mathcal{U}}\to{\mathbb{R}}$ such that $h(x_{0})=y_{0}$ and for all
$x\in{\mathcal{U}}$, $H(x,h(x))=0$. Moreover, for any $x\in{\mathcal{U}}$,

$\displaystyle\frac{\partial}{\partial x}h(x)$
$\displaystyle=-\frac{\frac{\partial}{\partial
x}H(x,h(x))}{\frac{\partial}{\partial y}H(x,h(x))}.$

######  Proposition G.6 (Davis–Kahan [DK70]).

Let $A,B\in{\mathbb{R}}^{d\times d}$ be symmetric matrices. Then

$\displaystyle\min\left\\{\left\|v_{1}(A)-v_{1}(B)\right\|_{2},\left\|v_{1}(A)+v_{1}(B)\right\|_{2}\right\\}$
$\displaystyle\leq\frac{4\left\|A-B\right\|_{2}}{\max\left\\{\lambda_{1}(A)-\lambda_{2}(A),\lambda_{1}(B)-\lambda_{2}(B)\right\\}}.$

Note that the minimum on the left-hand side is to resolve the sign ambiguity
since $v$ is an eigenvector if and only if $-v$ is.


# The optical simulation model of the DarkSide-20k Veto detector

Cenk Türkoğlu and Sarthak Choudhary for the DarkSide collaboration  AstroCeNT,
Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, Rektorska
4, 00-614 Warsaw, Poland<EMAIL_ADDRESS>
###### Abstract

DarkSide-20k is a rare-event search experiment aimed at finding signals of
dark matter particles. It is a dual-phase detector that registers ionisation
and scintillation signals originating from the particles interacting with the
liquid argon detector medium. It is enclosed in a single-phase liquid argon
neutron veto tank, equipped with Gd-loaded panels for capturing neutrons.
Since vetoing and particle identification are carried out using the light
signal, it is crucial to maximise the light yield. Light collection efficiency
depends on optical properties of the detector and particularly for the veto
detector, which has a photosensor coverage of the order of a per cent, the
reflectivity of the walls has a big impact. To quantify the amount of
collected light, a comprehensive Geant4 simulation is performed, which uses
optical characterisation data. In this work, a detailed description of the
optics model for the veto of the experiment will be discussed.

##  1 Introduction

DarkSide-20k (DS-20k) is a dual-phase rare-event search experiment dedicated
to finding the signals of dark matter particles. To find this signal, DS-20k
employs liquid argon (LAr) as the detector medium, and uses ionisation and
scintillation-light signals to detect particle interactions. LAr is used in
the inner time projection chamber (TPC) ($\sim$ 50 tons) instrumented with
SiPMs which is enclosed in an active LAr veto, also enclosed in a passive LAr
shield tank [1]. The veto of the DS-20k is crucial for detecting neutrons that
can mimic dark matter induced signals. Neutrons are moderated by collisions in
the acrylic. Gd in the acrylic ensures the emission of multiple high energy
(with a threshold of $\mathrm{{}^{39}Ar}$ endpoint to define hits)
$\gamma$-rays after the neutron capture. For both scintillation and ionisation
channels particle detection is carried out via light signals, light collection
becomes an essential part for both the TPC and the veto.

(a)

(b)

Figure 1: (1a) DarkSide-20k inside the ProtoDUNE cryostat [1] and (1b) layers
of the veto made up of LAr, ESR and PEN

##  2 Simulation

In order to test the performance of the DS-20k veto and optimise it before
finalising the design, a Geant4 simulation code ”g4ds” [2] was developed. The
simulation code encapsulates a detailed geometry of the DS-20k detector and
the physics processes that are expected to occur in the experiment. The
simulation includes the geometry to the TPC, of the veto, and of the shield
tank. One of the most important aspects of the veto simulation is the optics
since the light collection efficiency can change dramatically depending on the
materials used and the design. For this reason, a detailed optical model was
implemented in the simulation code.

SiPMs are not sensitive to vacuum ultraviolet (VUV) scintillation light
emitted from particle interactions with LAr. Hence, wavelength shifter (WLS)
materials that can convert the wavelength of the incoming VUV photons into
optical photons are used both inside the TPC and in the veto. While in the
TPC, a vacuum evaporated tetraphenyl-butadiene (TPB) coating is used as the
WLS of choice, in the veto, polyethylene naphthalate (PEN) films are used.
Both of these materials have advantages with respect to each other [6]. Since
PEN is in the form of a thin film, various optical parameters should be taken
into account and used in the simulations. In addition to the WLSs, design and
materials used in the setup have parameters that can affect the light
collection.

###  2.1 WLS Properties

Since PEN is available as a large format film, it is much more scalable
compared to TPB, which makes it attractive for DS-20k and future liquid argon
based detectors. There various optical properties of PEN which are crucial to
the light collection in the veto.

One of the most significant parameters for WLSs is the wavelength shifting
efficiency. In the beginning of the design process for the DS-20k, TPB was
planned to be used as the WLS for both the veto and the TPC. However,
considering the advantages of using PEN, as well as the latest results from
one-to-one comparison of TPB and PEN in cryogenic conditions [6] lead to the
decision of using PEN as the WLS for the veto. In [6], the relative wavelength
shifting efficiency (WLSE) of PEN compared to that of TPB was reported to be
47.2 $\%$.

In the simulation absorption length for PEN was given separate values for VUV
and visible regimes. The two being separated roughly at 150 nm. The same
scheme was followed for Rayleigh scattering length as well. Absorption length
for the visible part of the spectrum, $\sim$1 cm [8]. For Rayleigh scattering
length, 150 $\mathrm{\mu m}$ is assumed for the visible part of the spectrum.
Refractive indices for PEN both for the visible and the UV parts are taken as
1.75. Emission spectrum of PEN, another parameter taken into account, can be
seen on Figure 2a. Re-emission time used in the simulations was observed to be
20 ns [4] for PEN.

###  2.2 Properties of the Veto Materials

In addition to the parameters related to the WLSs, optical properties of other
materials in the veto are also essential in the light collection. One of these
properties is the absorption length of LAr, which is assumed 100 m for both
visible and UV part of the spectrum.

Reflectivities of other parts of the veto are crucial as even a minuscule
change can change the light yield (LY). The walls of the veto detector are
covered with a thin film called Enhanced Specular Reflector (ESR), which
enhances the light collection of the SiPMs with its $~{}98~{}\%$ reflectivity.
Reflectivity curve of ESR is shown on Figure 2b in black dashed line. Another
material with reflectivity information used in the simulation is the SiPMs.
The reflectivity of the SiPM surface indirectly affects the amount of photons
detected (see green dashed line on Figure 2b), as well as the photon detection
efficiency (PDE) of the SiPM [3].

(a)

(b)

Figure 2: Spectra of various optical parameters used in the simulation: (2a)
emission spectra of PEN at room temperature (red) and at cryogenic temperature
(blue) [5] and (2b) reflectivity percentages of ESR (black), PEN air-coupled
to ESR (blue) and SiPM (green) [6]

##  3 Results and Conclusions

After inserting the parameters discussed in sections 2.1 and 2.2, simulations
were performed. Individual effects of these parameters were investigated in
order to understand the strength of their effects on the LY. To understand
whether the final results are correct or not, a comparison with a simple
analytical model was made [7]:

$\mathrm{LY=40\>[ph/keV]\cdot PDE\>[pe/ph]\cdot WLSE\cdot\frac{F_{sens}\cdot
FF\cdot(1-R_{sens})}{1-(F_{sens}\cdot R_{sens}+(1-F_{sens})\cdot R_{wall})}}$
(1)

This model takes into account the wall reflectivity ($\mathrm{R_{wall}}$), the
SiPM coverage fraction ($\mathrm{F_{sens}}$), the SiPM reflectivity
($\mathrm{R_{sens}}$), the SiPM PDE (PDE), fill factor (FF) and the WLSE.

The Geant4 simulation was validated against the analytical model, and the
results can be seen on Figure 3. The graph shows the LY in units of
photoelectrons (pe) per energy (keV), and it is drawn with respect to the
photosensor (SiPM, in the case of DS-20k) coverage fraction, which is the
trend future dark matter direct detection experiments are working towards.
This comparison confirmed the suitability of PEN as WLS to be used in the
DS-20k veto, despite its lower WLSE compared to that of TPB.

Figure 3: Light yield output of the veto simulations (red) with respect to
different configurations of the SiPM coverage fraction compared to the
analytical model (black) according to [7]

.

The DarkSide Collaboration would like to thank LNGS and its staff for
invaluable technical and logistical support. This report is based upon work
supported by the U. S. National Science Foundation (NSF) (Grants No.
PHY-0919363, No. PHY-1004054, No. PHY-1004072, No. PHY-1242585, No.
PHY-1314483, No. PHY- 1314507, associated collaborative grants, No.
PHY-1211308, No. PHY-1314501, No. PHY-1455351 and No. PHY-1606912, as well as
Major Research Instrumentation Grant No. MRI-1429544), the Italian Istituto
Nazionale di Fisica Nucleare (Grants from Italian Ministero dell’Istruzione,
Università, e Ricerca Progetto Premiale 2013 and Commissione Scientific
Nazionale II), the Natural Sciences and Engineering Research Council of
Canada, SNOLAB, and the Arthur B. McDonald Canadian Astroparticle Physics
Research Institute. We acknowledge the financial support by LabEx UnivEarthS
(ANR-10-LABX-0023 and ANR-18-IDEX-0001), the São Paulo Research Foundation
(Grant FAPESP-2017/26238-4), and the Russian Science Foundation Grant No.
16-12-10369. The authors were also supported by the “Unidad de Excelencia
María de Maeztu: CIEMAT - Física de partículas” (Grant MDM2015-0509), the
Polish National Science Centre (Grant No. UMO-2019/33/B/ST2/02884), the
Foundation for Polish Science (Grant No. TEAM/2016-2/17), the International
Research Agenda Programme AstroCeNT (Grant No. MAB/2018/7) funded by the
Foundation for Polish Science from the European Regional Development Fund, the
European Union’s Horizon 2020 research and innovation programme (DarkWave,
Grant No. 962480), the Science and Technology Facilities Council, part of the
United Kingdom Research and Innovation, and The Royal Society (United
Kingdom). I.F.M.A is supported in part by Conselho Nacional de Desenvolvimento
Científico e Tecnológico (CNPq). We also wish to acknowledge the support from
Pacific Northwest National Laboratory, which is operated by Battelle for the
U.S. Department of Energy under Contract No. DE-AC05-76RL01830.

## References

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