# Using Human-Guided Causal Knowledge for More Generalized Robot Task Planning

Semir Tatlidil,1 Yanqi Liu,2 Emily Sheetz,3 R. Iris Bahar,2 Steven Sloman,1

###### Abstract

A major challenge in research involving artificial intelligence (AI) is the
development of algorithms that can find solutions to problems that can
generalize to different environments and tasks. Unlike AI, humans are adept at
finding solutions that can transfer. We hypothesize this is because their
solutions are informed by causal models. We propose to use human-guided causal
knowledge to help robots find solutions that can generalize to a new
environment. We develop and test the feasibility of a language interface that
naïve participants can use to communicate these causal models to a planner. We
find preliminary evidence that participants are able to use our interface and
generate causal models that achieve near-generalization. We outline an
experiment aimed at testing far-generalization using our interface and
describe our longer terms goals for these causal models.

## Introduction

One of the most impressive feats of human intelligence is our ability to
transfer learned knowledge to solve a problem in a novel environment (Shepard
1987). Despite their overall success, state-of-the-art machine learning
methods struggle with this (Edmonds et al. 2020; Nair et al. 2019).

What makes this problem challenging is determining an appropriate level of
granularity for state representations. If an agent represents the states of
its environment at a very detailed level, then it will likely fail to apply
the solution that it learned in a new environment because even small changes
in the environment will appear as new states. For example, consider the task
of putting together an object that can emit light. In the training
environment, we learn that connecting wax and a wick and then lighting the
wick will emit light. If we try to apply this knowledge in a new environment
where wax is replaced with a container filled with oil, we can only do this if
we know that wax and oil share some critical feature that makes them interact
with the wick in a similar way to emit light. In other words, we need to
construct an abstract model of the states in our environment.

The importance of constructing abstract state representations for human-like
intelligence has been emphasized by AI researchers (Konidaris 2019). Previous
works has shown that abstract state representations can be learned
automatically from simulations using reinforcement learning (RL) (Abel,
Hershkowitz, and Littman 2016) or generative adversarial nets (Kurutach et al.
2018). Studies focusing on transfer learning show that abstract
representations lead to more efficient solutions in novel tasks (Asadi, Abel,
and Littman 2020; Kokel et al. 2021). However, a major drawback of these
methods is that they may require many trials in order to learn a good abstract
state representation, and the learned representations may not be interpretable
by humans.

In contrast to AI research, studies with people suggest that they are able to
construct abstract representations with limited experience and are able to
generalize this knowledge across tasks (Holyoak 2012). What makes people so
efficient in constructing abstract representations? One line of evidence
suggests that it is people’s tendency to think about events in causal terms.
Studies looking at how people create categories, which can be seen as a method
of state abstraction by aggregating instances, show that people are heavily
influenced by causal information in category formation (Marsh and Ahn 2009;
Waldmann and Hagmayer 2006). Moreover, they tend to employ these causal
categories to solve problems of a similar nature in new environments (Lien and
Cheng 2000). In complex domains where a multitude of features can be used for
categorization (such as categorizing different types of fish), evidence
indicates that novices are more likely to categorize by considering surface
level features such as perceptual similarity. On the other hand, experts are
more likely to categorize based on causal information (Rottman, Gentner, and
Goldwater 2012; Shafto and Coley 2003). Overall, these studies provide strong
evidence that humans tend to create abstract categories using causal
information and use such information in a variety of (simple or complex)
tasks.

The central role of causality in achieving human-like intelligence has been
recognized by AI researchers particularly with respect to learning abstract,
generalizable knowledge (Schölkopf et al. 2021; Zhu et al. 2020). For
instance, Edmonds et al. (2020) investigated whether learning causal
information in an escape room task would lead to better performance in a new
environment. They showed that while a number of RL agents showed little to no
transfer of knowledge across environments, an agent that learned abstract
causal rules was able to transfer this knowledge to the new environment and
learn the new solution much quicker. Moreover, the behavior of the causal
agent was similar to the behavior of human subjects. This supports both the
importance of causality in human thinking and how causal knowledge aids in
generalizing knowledge across environments. In a different task that involved
learning the relationship between many electrical switches and light bulbs,
Nair et al. (2019) have found that an agent that learns causal rules is better
able to use this knowledge to solve tasks in new environments compared to
model-free RL agents that do not learn the causal rules. Overall, the evidence
strongly suggests that understanding the causal mechanisms of a task is
critical for generalizing performance to new environments both in human agents
and in AI.

Our goal is to develop a human-robot interaction system where causal knowledge
communicated by a human is used by the robot to complete specific tasks and
apply this knowledge to novel tasks. We propose to use causal information
communicated to the robot by a human in the form of causal graphical models
(Sloman 2005). These causal models will communicate which features of the
environment are relevant to the task, giving us an appropriate state
abstraction. To the extent that a new task fits into the same causal model,
this knowledge can be reused.

In this paper, we present the first step of this work where we develop a user
interface that allows people to communicate causal models of how various
objects work to produce light. We conducted a preliminary online experiment to
test if naïve users could use our interface to express their causal models and
to see if their causal models would achieve near-generalization.

## Assembling objects using causal models

Our approach can be described in three steps. First, participants use an
interface to express the causal model of how an object works to produce light.
The users are required to express a causal graph in terms of what function
each object part performs that causes light to be emitted, instead of
describing the object parts themselves. Second, we interpret the user-
generated causal graph as a reward function for a high-level symbolic planner.
The planner generates a sequence of actions to assemble the object. Finally,
the high-level symbolic planner will be connected to a simulator that converts
the plan into a sequence of motor commands and visually demonstrates the
execution of the task. In our experiment, we have four light-producing
objects: a candle, a kerosene lamp, a desk lamp, and a flashlight. A candle
and a kerosene lamp are considered to be from the same category of ”light
production method” as they both burn fuel to produce light. A desk lamp and a
flashlight are considered to be from the same category as they both use
electricity to produce light.

Our definition of a successful causal model is a causal model that can be used
by the planner to generate an accurate plan to assemble the new object,
regardless of the specific labels chosen by the participants to represent the
functions of the object parts. If the causal model of an object can be used to
generate a plan for another object from the same category, we consider this a
successful near-generalization. If it can be used to generate a plan for an
object from a different category, then it is a successful far-generalization
as shown in Fig.1. In this experiment, we only tested near-generalization.

Training object |  Near-generalization |  Far-generalization
---|---|---
|  |
Desk lamp |  Flashlight |  Candle

Figure 1:  Near- and far-generalization

## Near-generalization experiment

Nine undergraduates attending Brown University participated for course credit.
They first read a short tutorial explaining that their task was to describe
how an object produced a certain effect by constructing the causal model using
our interface. The tutorial explained how the interface worked with an example
object that was different than the light-producing objects. They were then
directed to an experiment page that involved three steps. In the first step,
they were presented with one of the four light-producing objects (order was
counterbalanced across participants) with a short explanation of how the
object worked, as well as a picture of the object with labeled parts. They
identified the functions of object parts relevant to achieving the final
effect - light. They did this by writing down function labels for object parts
themselves as shown in Fig.2. They were allowed to associate multiple
functions with a single part, and multiple parts with a single function.

The second step involves describing the causal model of the object in terms of
the functions of the object parts. The causal model is a collection of causal
rules that are expressed using predefined keywords. Each rule establishes
which function or multiple functions of object parts cause an effect. The
final effect is always the goal (i.e., light), but participants were allowed
to enter intermediate effects (that were different from functions of object
parts identified in the first step) that can be used as causes in another
causal rule. After they entered their causal model, they saw their model in
graphical format and the interface generated a plan using their causal model
and presented it to them in a text format describing which object parts are
connected as shown in Fig.3. If the generated plan was not satisfactory,
participants could go back and update their causal models.

Figure 2: Step 1 user input example

The presentation of the plan is intended to establish a dynamic communication
channel between the human and the robot: the robot uses the causal model to
generate a plan and the human sees how this causal model affects the robot’s
planning behavior, which can help the human understand what needs to be
changed in the causal model if the plan is not satisfactory.

Figure 3: Causal model created by a user in Step 2 for the desk lamp.
Functions of the object parts identified by the same user are shown in Fig.2

In the final step, we tested for near-generalization by presenting a new
object from the same category as the training object and asked the
participants to associate the functions of the new object parts with the
functions they described for the previous object. Then, the planner tried to
create a plan for the test object using the causal model that was created for
the training object. Note that participants were not allowed to create a new
causal model for the test object or to update their causal model of the
training object. As a result, the planner could only succeed if the causal
model of the training object was abstract enough to describe how the test
object worked. Each participant went through this 3-step process once for each
category.

Figure 4: Step 3 user input example

### Planner

The planner is in charge of generating a sequence of actions. The actions that
a planner generates are high-level action primitives such as, screw, insert,
connect. The high-level action primitives will be processed to generate motor
commands that can be performed by a simulator or a real robot. We choose to
use Markov Decision Process (MDP) as the framework for planning. MDP captures
the probability of state transition when an action is performed, which
accounts for robot action uncertainty and object parts compatibility. The MDP
problem is defined by $\\{S,A,T,R\\}$, where $S$ represents the current
assembly state (i.e., the current object parts assembled, their positions in
the assembled structure), $A$ is the set of predefined actions, $T$ is the
transition probability, and $R$ is the reward function at a given state. The
transition probability $T=P(S,A,S^{\prime})$ can be estimated by the
compatibility of the connection points between two object parts $O_{1}$,
$O_{2}$. The connection compatibility can be pre-defined with binary values or
can be estimated based on the geometric alignment of the connectors.

For the reward function $R$, we use the causal model $G$ created by the user
to guide the reward for each state. The causal model generates a score
representing whether a given state is able to achieve the final effect. Each
node $V$ in the causal graph is a binary value. For a current assembled
structure with object $O_{1}$ .. $O_{n}$, function nodes associated with each
object parts will have value a of 1. The value of the final effect node,
$V_{goal}$, is determined by a functional relationship $f$ of the values of
its parent nodes and calculated recursively until the function nodes (i.e.,
root nodes) are reached:

$V_{goal}=f(parent(V_{goal}))$ (1)

The reward function is dependent on the value of $V_{goal}$. We assign
positive reward if the goal is reached and negative reward if goal is not
reached and there is no further applicable actions.

## Results

The results of our preliminary experiment suggest that most people were able
to use our interface to express reasonable causal models with minimal
instructions that they received on the tutorial page. An example of a
successful model created by a participant for the desk lamp (training) that
transferred to the flashlight (test) is shown in Fig.3 and Fig.4.

This provides preliminary evidence supporting the claim that causal models
created by people can achieve near-generalization. Moreover, it also shows
that our interface can be used to communicate these causal models. In some
cases, participants created valid causal models for the training object but
their model failed to transfer to the test object. An example model that
failed to transfer from desk lamp to flashlight is shown in Fig.5. This model
fails because of two reasons. First, the participant did not specify any of
the flashlight parts with the function “turn electricity into light” that was
in their causal model, and as such, the flashlight parts were not enough to
achieve the goal. Second, in step 3 the user added a new function “hold things
together” for the “case” part of the flashlight that was not present in their
earlier causal model.

Some of the failures of transfer are likely due to participants
misunderstanding the task. For instance, it is likely that the participant
decided to include “hold things together” as a function for the flashlight
because they misunderstood the task to be describing the functions of all
object parts instead of describing only the functions relevant to producing
light. The same participant included “diffuse light” function of the “shade”
of the desk lamp to be necessary for achieving the goal of producing light,
which clearly cannot be true since one has to produce light before it can be
diffused. In the next experiment, we will clarify subtle differences like
these to elicit more appropriate causal models for our task.

desk lamp parts | functions
---|---
shade | diffuse light
light bulb | turn electricity into light
base with cables | provide electricity
flashlight parts | functions
case | hold things together
head | diffuse light, provide electricity
batteries | provide electricity

Figure 5: Example of a failed causal model generated by a user. Table on the
top shows the step 1 input for the training object (desk lamp) and the table
below shows the step 3 input for the test object (flashlight)

Finally, in a few trials participants generated invalid causal models. For
instance, some users specified intermediate effect nodes in their causal
models but did not specify their causes despite being instructed to do so in
the tutorial. We will prevent the users from submitting such causal models and
present them with an error message explaining what they need to change.

## Far-generalization experiment

For our next experiment, we will update our interface to address the
ambiguities that led some participants to generate causal models that were
invalid or failed to transfer, as discussed in the previous section. We will
use the updated interface to test for both near-generalization and far-
generalization. We will present two training objects to participants and ask
them to generate a single causal model that describes how they produce light.
Half of the participants will be assigned to a near-generalization condition
where the two training objects they see will be from the same category (i.e.,
producing light by burning fuel or by using electricity). The other half will
be in a far-generalization condition and they will see one object from each
category. The test phase will be identical for both groups of participants:
they will see a novel object from one category and will be asked to complete
step 3 of the current experiment as described in the ”Near-generalization
experiment” section. Then, they will see a second test object from a different
category than the first test object and will be asked to repeat step 3 with
this object as well. We predict that presenting two objects from the same
category will lead participants to generate causal models that will only
generalize to the test object from the same category, so participants in the
near-generalization condition are predicted to create models that will achieve
near-generalization but not far-generalization. In contrast, presenting two
objects from different categories to the participants in the far-
generalization category should lead them to think in even more abstract terms,
so we predict that their causal models will generalize to both test objects.
The order of objects presented as training and test objects will be
counterbalanced across participants.

## Future Work

In addition to our next experiment, in the future, we are planning to use an
online platform, such as Amazon Mechanical Turk, to recruit a high number of
participants and test the generalizability of the causal models they generate.
Inspired by recent developments in relation networks (Sung et al. 2017) and
graph neural networks (Zhou et al. 2018), we aim to use a neural network to
extract a single causal model from a large sample of user-generated causal
models. We predict that a causal model learned from a large sample of users is
more likely to be accurate compared to a model generated by an average user,
given the evidence suggesting that collective decision making tends to
outperform decisions of individuals (Surowiecki 2005) even if the individuals
are experts (Wolf et al. 2015).

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# Frequency stability of cryogenic silicon cavities with semiconductor
crystalline coatings

Dhruv Kedar  1,* Jialiang Yu  2 Eric Oelker  1,3 Alexander Staron  1 William
R. Milner  1 John M. Robinson  1 Thomas Legero  2 Fritz Riehle  2 Uwe Sterr  2
and Jun Ye1  1JILA, NIST and University of Colorado, 440 UCB, Boulder,
Colorado 80309, USA
2Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig,
Germany
3University of Glasgow, Institute for Gravitational Research, School of
Physics and Astronomy, Glasgow G12 8QQ, United Kingdom
<EMAIL_ADDRESS>
###### Abstract

State-of-the-art optical oscillators employing cryogenic reference cavities
are limited in performance by the Brownian thermal noise associated with the
mechanical dissipation of the mirror coatings. Recently, crystalline
Al1-xGaxAs/GaAs coatings have emerged as a promising candidate for improved
coating thermal noise. We present measurements of the frequency noise of two
fully crystalline cryogenic reference cavities with Al0.92Ga0.08As/GaAs
optical coatings. We report on previously unmeasured birefringent noise
associated with anti-correlated frequency fluctuations between the
polarization modes of the crystalline coatings, and identify variables that
affect its magnitude. Comparing the birefringent noise between the two
cryogenic reference cavities reveals a phenomenological set of scalings with
intracavity power and mode area. We implement an interrogation scheme that
cancels this noise by simultaneous probing of both polarization modes. The
residual noise remaining after this cancellation is larger than both cavities
thermal noise limits, but still lower than the instabilities previously
measured on equivalent resonators with dielectric coatings. Though the source
of these novel noise mechanisms is unclear, we demonstrate that crystalline
coatings can provide stability and sensitivity competitive with resonators
employing dielectric coatings.

††journal: optica††articletype: Research Article

##  1 Introduction

Ultra-stable lasers locked to cryogenic reference cavities represent an
indispensable tool for rapid characterization of optical atomic clocks [1, 2],
development of optical timescales [3, 4], and tabletop investigations of
fundamental physics [5, 6, 7]. The performance of state-of-the-art optical
oscillators is fundamentally limited by length fluctuations associated with
thermally driven mechanical dissipation within the mirror coatings [8, 9, 10].
Thus, low mechanical loss coatings with high reflectivity operating at low
temperatures are desirable for improving the performance of these optical
reference cavities [11, 12, 13, 14] and interferometric gravitational wave
detectors [15, 16, 17, 18].

Studies of crystalline mirror coatings have revealed several materials that
exhibit lower mechanical loss than their traditionally-used amorphous
dielectric counterparts [19, 20]. Room temperature loss angles of crystalline
Al0.92Ga0.08As/GaAs ($\phi=2.5\times 10^{-5}$ [21]) are measured to be over
ten times smaller than those of amorphous SiO2/Ta2O5 layers (4 K loss angle of
$5.6\times 10^{-4}$ [22]). However, the majority of studies have extracted the
coating loss angle via ringdown measurements of mechanical cantilevers [23,
24], far from the frequency band of operation. Direct measurements of coating
noise [25] have been less frequently studied, but can provide broadband
characterization of thermal noise and any other mechanisms that can generate
sources of length noise. Interferometric measurements of coating noise are
therefore imperative to the operation of gravitational wave detectors [22].
Furthermore, the associated length noise and loss angles have yet to be
characterized in a cryogenic resonator system at the audio band and below.
Cryogenic silicon cavities have been established as well-characterized and
powerful platforms for studying the length noise associated with SiO2/Ta2O5
dielectric optical coatings [26, 12]. In this letter, we present the
characterization of crystalline Al0.92Ga0.08As/GaAs coatings transferred onto
silicon substrates and operated on a 6 cm monocrystalline silicon cavity
spacer at temperatures of 4 K and 16 K, referred to as Si6. Using these
coatings and assuming that the measured room temperature loss angle persists
at our operating temperatures [25], we expect a fractional frequency
instability of $1.3\times 10^{-17}$ at a cavity temperature of 4 K. A
comparison is made to a previous system Si4 (Fig.1), an equivalent cavity with
dielectric SiO2/Ta2O5 coatings detailed in [13, 14].

Investigations of the Si6 system reveal frequency noise associated with the
large birefringent mode splitting of the crystalline mirror coatings. We
identify an optical intensity scaling of this birefringent noise and compare
it to measurements made on a similar 21 cm silicon resonator with crystalline
coatings operated at 124 K (Si5). We suppress this noise by employing a novel
simultaneous dual-tone probing of the two polarization eigenmodes of the
cavity [27]. The improved cavity stability on both systems is then limited by
a residual noise of unknown origin, significantly above the expected thermal
noise limit calculated using the room temperature loss angle of
Al0.92Ga0.08As/GaAs.

Figure 1: Coating thermal noise contributions ($\approx
S_{\textrm{Brownian}}$) for a variety of cryogenic silicon cavities referenced
and compared in this work, with labels detailing spacer length, operating
temperature, and optical coating material. The coating label "dielectric"
specifically refers to SiO2/Ta2O5, and "crystalline" refers to
Al0.92Ga0.08As/GaAs. For thermal noise calculations of Si5 and Si6, we assume
the room temperature value for the Al0.92Ga0.08As/GaAs loss angle [25] is
maintained at cryogenic temperatures. Conversely, Si3 and Si4 achieved thermal
noise limited performance at these levels [1, 14, 22].

##  2 Methods

Though the spacer and substrate parameters of Si4 and Si6 are identical,
crystalline coatings introduce some key differences and complexities. The
mirror coatings are composed of an 11.6 $\mu$m-thick multilayer stack of
alternating high-index GaAs and low-index Al0.92Ga0.08As, and are oriented
with parallel coating polarizations [28]. The GaAs $\langle 100\rangle$ axis
is normal to the coating, and parallel to the crystalline $\langle 111\rangle$
axis of the crystalline silicon. Two near-planar mirrors of 1 m radius of
curvature realize cryogenic transmission ($T$) of 11.8 ppm and loss ($L$) of
10 ppm combined for both mirrors, supporting a finesse of 290,000. We measure
a $770\textrm{ kHz}$ birefringent splitting of the polarization modes, well
resolved outside of the $8.6\textrm{ kHz}$ cavity linewidth. The cavity is
housed in a closed-cycle commercial cryostation that enables cooling the
cavity down to a minimum temperature of 4.7 K.

We interrogate the cavity with a unique measurement scheme that enables
reduction of the frequency noise associated with the birefringent mode
splitting. The optical layout is illustrated in Fig.2(a). An error signal is
derived from the cavity reflection by probing in the traditional Pound-Drever-
Hall (PDH) technique, with modulation sidebands at 5 MHz generated via a fiber
coupled electro-optic modulator (EOM). A portion of this signal is picked off
before the cavity for active cancellation of residual amplitude modulation
(RAM) with a control signal applied as a bias voltage to the EOM [29]. We
stabilize the optical power in cavity transmission with feedback to an
acousto-optic modulator (AOM1). Conventional PDH schemes address a single
polarization mode of the resonator. This does not allow for any cancellation
of birefringent noise, though it has not appeared to be a significant source
of instability in cavities with dielectric coatings. Unique to this setup is
the addition of AOM2 which synthesizes two frequency tones 770 kHz apart that
can be used to address both polarization modes of the cavity. Both tones are
imprinted with a corresponding set of phase modulation sidebands as
illustrated in Fig.2(b). A half waveplate before the cavity is set to couple
an equal intensity of light into both polarization modes while maintaining
equivalent discrimination slopes for their error signals. The detected signal
on the PDH photodetector (PD) is the sum error signal of both polarization
modes and any anti-correlated noise between the two is significantly
suppressed. We routinely ensure that optical powers and discrimination slopes
are matched to within $3\%$ between the two modes, and we project that this
enables $>20$ dB rejection of any anti-correlated signal. We note that RAM
stabilization actuates on the sum error signals for both polarizations as
well. Intensity stabilization suppresses any fluctuations in transmitted power
for both optical modes. Si5 implements a different scheme for birefringent
noise cancellation, detailed in [30].

A compilation of characterized noise sources is detailed in Fig.3(a).
Technical noise contributions are evaluated by propagating the ambient
environmental noise through a measured transfer function. Significant upgrades
were implemented to enable observation of the Si6 thermal noise. Noise
contributions from electronic sources were reduced. The cavity mount was
redesigned as a stainless steel tripod, and we identified a sinusoidal
dependence of the horizontal acceleration sensitivities
$\kappa_{x},\kappa_{y}$ that oscillates with the azimuthal angle formed
between the cavity $\langle 10-1\rangle$ axis and the tripod support.
Sinusoidal dependence of the horizontal acceleration sensitivity varies out of
phase with the vertical sensitivity $\kappa_{z}$, and the operating angle is
chosen to minimize the total vibration noise coupling to the cavity length.
Ultimately we realize a 10-fold improvement over Si4 of horizontal
acceleration sensitivity to values of $\kappa_{x}=1\times
10^{-11}\textrm{g}^{-1}$ and $\kappa_{y}=2\times 10^{-11}\textrm{g}^{-1}$,
with vertical sensitivity $\kappa_{z}=3\times 10^{-11}\textrm{g}^{-1}$ near
DC.

Figure 2: (a) Optical layout for probing the Si6 system with active RAM
suppression at a modulation frequency (5 MHz) where the cavity error signal is
encoded in a traditional PDH locking setup. Dual-tone probing is achieved by
driving AOM2 with two RF tones separated by the birefringent mode splitting of
Si6 (green and purple sets of tones). A half waveplate before the cavity is
aligned to couple an equal mixture of s- and p-polarized light to the two
polarization modes of the cavity, labeled $\hat{E}_{1}$ and $\hat{E}_{2}$. The
collinear beams are then launched upwards with a 45∘ mirror to couple into the
cavity. The reflected cavity error signal has equal contributions from noise
of the two polarization modes, and any anti-correlated noise is automatically
rejected. (b) Frequency landscape of the tones applied to the Si6 cavity. The
two tones applied to AOM2 separate the carrier and the two sidebands each into
two tones separated by the birefringent mode splitting of the cavity. The
purple tones shown address one polarization mode ($\hat{E}_{1}$) while the
green tones simultaneously address the orthogonal mode ($\hat{E}_{2}$).

The thermal environment of the cryostation, while conceptually similar to
[13], is also overhauled to support long term stability of projected Si6
thermal noise. All thermal shields are reconstructed out of high purity
oxygen-free copper and gold-plated to reduce radiative coupling. The thermal
isolation stages are engineered as a series of cascaded low pass filters.
Adding holmium copper after the stage 2 coldfinger and integrating thermally
resistive carbon fiber reinforced plastics into the support structure have
increased the thermal isolation between cryogenic stages. The thermal time
constant between active and passive 4 K thermal shields is increased nine-fold
to 12000 s. The redesigned cavity support structure results in a reduced
passive shield-cavity time constant of 600 s, down from 1200 s, despite
superior vibration performance.

From Fig.3(a), it is clear that RAM dominates all other measured noise
sources. We employ active RAM cancellation but are limited by out of loop
parasitic effects occurring after the stabilization point (RAM PD in
Fig.2(a)). Aligning to the $\textrm{HG}_{00}$ cavity mode introduced an etalon
with the input vacuum optic, and the etalon effect was strongly reduced by
altering the angle of incidence to couple the $\textrm{HG}_{01}$ transverse
mode. The reduced mode coupling efficiency increases the shot noise
contribution, evident in the RAM signal.

Figure 3: (a) Technical noise contributions for Si6. The projected thermal
noise is displayed (dashed black line). (b) Si6 frequency instability measured
with single polarization probing (red trace, $P_{c}$ = 18 mW) and residual
noise, with birefringent noise suppressed via dual polarization probing (blue
trace). The red trace measures
$(S_{\textrm{Brownian}}+S_{\textrm{biref}}+S_{\textrm{residual}})$, while the
dual polarization lock only measures
$(S_{\textrm{Brownian}}+S_{\textrm{residual}})$. A sum of technical noises is
displayed as the grey trace.

Instabilities of Si6 are measured via direct beat over a noise-cancelled
optical fiber [31] with Si3, a 21 cm long silicon cavity with dielectric
coatings operated at 124 K [1, 12]. Knowledge of the reference laser’s noise
spectrum is crucial and the instability of the latter system is verified to be
thermal noise limited with fractional frequency noise power spectral density
$S_{y}=1.7\times 10^{-33}/f$ and fractional instability $\textrm{mod
}\sigma_{y}=3.5\times 10^{-17}$ from 1-1000 seconds [1]. The frequency noise
of Si6 is calculated by measuring the combined fractional frequency
instability of a Si3-Si6 optical heterodyne beatnote on a zero dead-time
lambda-type counter. The Si3 instability contribution is then subtracted off
in quadrature.

The other cryogenic silicon cavity employing Al0.92Ga0.08As/GaAs coatings
(Si5) is 21 cm long and operated at 124 K [32, 30]. Insight into the principle
quantities driving the birefringent noise can be gained by comparing Si6
measurements to Si5. Coatings on this cavity share the same physical
properties (layer stack, thickness) as those on Si6, though the optical
qualities are slightly superior. Si5 realizes a finesse of over 350,000
corresponding to optical transmission and loss of $T=11$ ppm, $L=6.4$ ppm
between both mirrors. The measured birefringent mode splitting of 205 kHz is
roughly proportionally smaller due to the 3.5 times increased spacer length
relative to Si6. Details of operation and technical noise characterization can
be found in [30].

##  3 Results

We adopt the formalism of [30] to describe the Brownian thermal noise,
birefringent noise, and residual noise contributions to the measured
instability of Si6:

$S_{\textrm{T}}=S_{\textrm{Brownian}}+S_{\textrm{biref}}+S_{\textrm{residual}}$
(1)

On the Si6 system, the single polarization lock revealed that birefringent
noise was significantly higher than all technical contributions (Fig.3(b)) as
well as the projected coating thermal noise. Additionally, this birefringent
frequency noise is observed to scale with the optical power coupled into the
cavity, where a larger optical power corresponds to a larger cavity
instability. The spectrum of this power dependent noise appears to exhibit a
frequency-dependent slope which consistently exceeds a slope of $1/f$ in our
measurement bandwidth. Below an intracavity power of 18 mW (100 nW measured in
transmission), we do not measure any further reduction of birefringent
frequency noise. We depict this with a dashed grey 1/f guide in Fig.5(a)
corresponding to $S_{y}=2\times 10^{-32}/f$.

We increase the cavity temperature to 16.7 K, where silicon exhibits a zero
crossing of the coefficient of thermal expansion [33]. Comparison with the
single polarization data taken at 4.7 K exhibits no difference, indicating
that there is no strong thermal dependence of this phenomena between these two
temperatures (Fig.5(a)).

Figure 4: Mode area effect on birefringent noise. We separately measure the
birefringent noise of the $\textrm{HG}_{00}$ and $\textrm{HG}_{01}$ modes by
addressing a single polarization of the cavity. The ratio of birefringent
noise $S_{\textrm{biref}}^{01}$ and $S_{\textrm{biref}}^{00}$ for the two
modes reveals the dependence on mode area. If the birefringent noise is
independent of the mode area, we expect a ratio of 1. We find a ratio of 0.71,
consistent with the expectation of 0.75 if the noise scales with mode area
$g_{mn}w_{0}^{2}$. A large peak visible at 0.057 mHz is the result of a ground
loop that appeared in the $S_{\textrm{biref}}^{01}$ dataset and was omitted in
the fit.

We also find that the optical mode area on the mirror plays a significant role
in the magnitude of the birefringent noise. Both cavities have interrogated
the fundamental and higher Hermite-Gauss spatial modes $\textrm{HG}_{00}$ and
$\textrm{HG}_{01}$ and find that $S_{\textrm{biref}}$ is reduced by the ratio
of these two mode areas. We adopt the scaling factor $g_{mn}$ as defined in
[34] for the reduction of coating thermal noise of higher order modes. When
interrogating non-fundamental spatial modes, coating Brownian noise will be
modified as $g_{mn}S_{\textrm{coating}}^{00}$ where
$S_{\textrm{coating}}^{00}$ denotes the coating thermal noise of the
$\textrm{HG}_{00}$ mode. For the higher $\textrm{HG}_{01}$ mode, the scaling
factor is calculated as $g_{01}=0.75$. In Fig.4 we measure the birefringent
noise of the $\textrm{HG}_{00}$ and $\textrm{HG}_{01}$ modes at the same
intracavity power and temperature by interrogating a single polarization in
the Si6 system. Calculating the ratio
$S_{\textrm{biref}}^{01}/S_{\textrm{biref}}^{00}$ will highlight any
dependence on mode area. We find a ratio of 0.71$\pm 0.02$, similar to the
expectation of $g_{01}/g_{00}=0.75$ for birefringent noise dependent on mode
area $w_{0}^{2}$.

Figure 5: (a) Birefringent noise measured with a single polarization lock
exhibits a dependence on intracavity power $P_{c}$. Decreasing the intracavity
power reduces frequency noise down to a limit of $P_{c}$ = 18 mW, where
further reductions of power no longer result in an improvement of the cavity
instability despite technical noise sources remaining well below the noise
level. Although the character of the noise deviates strongly from $1/f$ in
different frequency regimes, we include a bound of $2\times 10^{-32}/f$
(dashed gray line) that identifies the lowest instability attained with the
single polarization lock. (b) Power-dependent birefringent noise is also
measured on Si5 at 124 K. (c) We observe an empirical scaling of this noise by
$1/\sqrt{P_{c}}$ for both systems. Rescaling the length noise $S_{x}$ by this
quantity yields a phenomenologically universal birefringent noise. As an
example, we display Si6 data at 16 K with this rescaling implemented. (d) A
comparison of the birefringent noise across the two systems, over a total of
three different temperatures with four different intracavity powers. We
rescale the length noise measurements by the observed dependencies on
intracavity power and mode area and observe a universal behaviour of the
birefringent noise.

Qualitatively similar noise characteristics for single polarization
interrogation have been observed in Si5 [30]. In contrast to measurements on
Si6 where we infer the birefringent noise by observing the reduction of
frequency with dual tone probing, birefringent noise in Si5 is measured
directly by locking two independent lasers to the fast and slow polarization
axes and measuring the frequency noise of their beatnote. Whereas
$S_{\textrm{biref}}\approx S_{\textrm{residual}}$ in Si5,
$S_{\textrm{biref}}\gg S_{\textrm{residual}}$ in Si6, so the inferred
measurement of birefringent noise only contains a $25\%$ contribution of
residual noise. Fig.5(a) and (b) display fractional frequency noise
measurements made in both systems. Frequency noise $S_{y}$ is significantly
lower on the 21 cm system for a given intracavity power, but we note that the
length noise $S_{x}$ is the proper quantity to compare as this noise
originates from the coatings. The longer spacer length of Si5 divides down the
fractional length noise of the identical coatings, suggesting a possible
strategy for mitigating birefringent noise. Comparison of $S_{x}$ between the
two cavities provides significantly better agreement, identifying the
crystalline coatings as the source for this excess birefringent noise. On both
cavities we identify an empirical scaling of the birefringent with intracavity
power as $1/\sqrt{P_{c}}$. That is, less power coupled into the cavity results
in reduced length noise. Normalizing $S_{x}$ by this quantity (Fig.5(d))
produces a family of curves that appear to share some universal noise
characteristics.

The slope of this noise increasingly deviates from $1/f$ at lower frequencies.
We note that the behaviour of this noise exhibits a similarity to electro-
optic effects where the refractive index of a medium varies with an incident
electric field. For Si5, an equivalent optical power bound cannot be
identified. Birefringent noise at circulating powers of 127 mW is at a level
comparable to the residual noise, thus a significant reduction of intracavity
power provides a marginal reduction in measured instability. At this
circulating power of 127 mW, we identify a frequency noise bound of
$S_{y}=1.9\times 10^{-33}/f$ or $S_{x}=8.5\times 10^{-35}/f$. We stress that
$\sqrt{P_{c}}$ scalings are empirical. Optical power build-up differs slightly
between the two cavities and it is unclear whether circulating power or
absorbed optical power is the more important metric. Nevertheless, rescaling
by this quantity does appear to approximately capture the measured power
dependence.

Figure 6: (a) Residual noise $(S_{\textrm{residual}}+S_{\textrm{Brownian}})$
after cancellation of the birefringent noise at both 4 K and 16 K shows no
temperature dependence. Residual noise of Si5 is at a similar level. (b) Long
term instability of the residual noise of the two cavities, and birefringent
noise of Si6. Both cavities exhibit significantly reduced long term
instability when compared to their dielectric counterparts Si2 [12] and Si4
[14]. Si6 realizes a fractional frequency instability of $5.5\times 10^{-17}$,
lower than the thermal noise limited instability of Si4. The $3.5\times
10^{-17}$ fractional instability of Si5 matches the thermal limit of Si2.

Unlike most fundamental sources of length noise, our measured birefringent
noise does not appear to be a typical thermodynamically driven process.
Investigating the noise of a single polarization mode at 4 K, 16 K, and 124 K,
we do not identify a strong dependence on the temperature of the crystalline
coating though data at all temperatures shares a similar scaling with optical
power. Room temperature cavities employing crystalline coatings may therefore
hide detrimental birefringent effects as the coating thermal noise can exceed
birefringent noise for low intracavity power.

A significant reduction in frequency noise is observed when we address both
cavity polarization modes simultaneously by applying two RF tones to AOM2. We
infer that the net improvement in cavity instability is due to first order
cancellation of birefringent noise. The current level of instability is
attributed to an unknown source of residual noise and no longer exhibits the
intracavity power dependence observed with birefringence noise. This implies
that the power dependence is a feature closely tied to the birefringent noise,
and that the residual noise is not limited by imperfect cancellation of
birefringence. Additional measurements where we introduce a slight imperfect
cancellation of birefringence (i.e. improper matching of intracavity powers
and discrimination slopes for the two modes) further confirms this, as the
residual noise does not increase.

Similarly, we find no dependence of the residual noise on which cavity spatial
mode we interrogate. This contrasts to measurements of birefringent noise,
where we observed that length fluctuations in the polarization modes splitting
could be reduced by probing a larger area of the crystalline coating. Similar
to studies in [30], we can establish that the residual noise is global in the
sense that it is highly correlated over the length scale of the two spatial
modes. It is possible that the current level of $S_{\textrm{residual}}$ is
limited by an uncharacterized source of technical noise. In the frequency band
of 0.01-0.3 Hz, we identify an instability consistent with
$S_{y}=3.37(9)\times 10^{-33}/f$ from a continuous 30,000 second measurement
against Si3. This performance is slightly better than the thermal noise
limited instability measured on Si4 of $S_{y}=4.12(5)\times 10^{-33}/f$, and
the long term instability for $\tau>1000$ s significantly exceeds it. However,
we note that the current instability of Si6 does not vary between cavity
temperatures of 4.7 K and 16.7 K. This suggests that either coating thermal
noise is not the primary source of instability, or that there is a strong
temperature dependence of the AlGaAs/GaAs mechanical loss angle.

Given that we find no reduction in instability when interrogating the larger
HG01 spatial mode, we infer that this residual noise is not the coating
thermal noise limit. Using the formalism of [22] for calculating the coating
thermal noise limit, we therefore place constraints on the maximum value of
the coating loss angle $\phi$:

$\displaystyle\phi_{4.7\textrm{ K}}<4.1\times 10^{-4}$ (2)
$\displaystyle\phi_{16.7\textrm{ K}}<1.2\times 10^{-4}$ (3)

These bounds already imply that these crystalline coatings can yield a
sensitivity improvement over conventional dielectric coatings at 4.7 K, and a
significant improvement at 16 K.

##  4 Conclusion

We have presented frequency noise measurements of crystalline
Al0.92Ga0.08As/GaAs coatings measured on a 6 cm and 21 cm cryogenic silicon
resonator. While the AlGaAs mechanical loss angle promises potential for
improved coating thermal noise at 4 K, 16 K, and 124 K, we identify a novel
source of instability associated with the mirror birefringence. We demonstrate
that this birefringent noise is optical intensity dependent, does not depend
strongly on temperature, and can be cancelled to reveal a temperature
independent residual noise. Without cancellation, we note that the fractional
frequency contribution of this birefringent noise can be reduced by employing
longer resonators, but applications requiring large intracavity power may
still be sensitive to these effects. The observed dependence on mode area
suggests that increasing beam size can reduce the magnitude of the
polarization fluctuations. Understanding the origins of this birefringent
noise will be imperative for the development of low mechanical loss coatings
as reduction of coating thermal noise is a significant obstacle for improving
reference cavities and gravitational wave interferometers.

Acknowledgments The authors thank T. Brown and A. Ellzey for technical
assistance, and G. Cole and J. Hall for insightful discussions.

Funding This work is supported by NIST, DARPA, AFRL, National Science
Foundation Physics Frontier Center (NSF PHY-1734006), and Physikalisch-
Technische Bundesanstalt. Ji.Y, T.L, F.R, and U.S. acknowledge support from
Project 20FUN08 NEXTLASERS and from the Deutsche Forschungsge- meinschaft
under Strategy–EXC-2123 QuantumFrontiers, Project-ID 390837967; SFB 1227
DQmat, Project-ID 274200144.

Disclosures The authors declare no conflicts of interest.

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

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