# A NOVEL GPR-BASED PREDICTION MODEL FOR CYCLIC BACKBONE CURVES OF REINFORCED
CONCRETE SHEAR WALLS

Zeynep Tuna Deger, Ph.D. and Gulsen Taskin Kaya, Ph.D.
Istanbul Technical University
<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

Backbone curves are used to characterize nonlinear responses of structural
elements by simplifying the cyclic force – deformation relationships. Accurate
modeling of cyclic behavior can be achieved with a reliable backbone curve
model. In this paper, a novel machine learning-based model is proposed to
predict the backbone curve of reinforced concrete shear (structural) walls
based on key wall design properties. Reported experimental responses of a
detailed test database consisting of 384 reinforced concrete shear walls under
cyclic loading were utilized to predict seven critical points to define the
backbone curves, namely: shear at cracking point ($V$cr); shear and
displacement at yielding point ($V$y and $\delta$y); and peak shear force and
corresponding displacement ($V$max and $\delta$max); and ultimate displacement
and corresponding shear ($V$u and $\delta$u). The predictive models were
developed based on the Gaussian Process Regression method (GPR), which adopts
a non-parametric Bayesian approach. The ability of the proposed GPR-based
model to make accurate and robust estimations for the backbone curves was
validated based on unseen data using a hundred random sampling procedure. The
prediction accuracies (i.e., ratio of predicted/actual values) are close to
1.0, whereas the coefficient of determination (R2) values range between 0.90 –
0.97 for all backbone points. The proposed GPR-based backbone models are shown
to reflect cyclic behavior more accurately than the traditional methods,
therefore, they would serve the earthquake engineering community for better
evaluation of the seismic performance of existing buildings.

Keywords: nonlinear modeling, reinforced concrete shear walls, cyclic
behavior, backbone curves, machine learning, Gaussian Process Regression

##  1 Introduction

###  1.1 Background on Shear Wall Modeling

Reinforced concrete shear walls are commonly used to resist earthquake loading
to provide substantial lateral strength and stiffness. As they are designed to
undergo inelastic deformations under high-intensity ground shaking, it is
essential to understand the nonlinear behavior and reflect the hysteretic
responses in analytical models as accurately as possible. There are various
well-accepted modeling approaches in the literature (e.g. summarized in NIST
GCR 17-917-45 [1]), that can be classified into two main approaches, namely:
finite element and member modeling. The finite element modeling approach
includes a detailed establishment of the problem by typically describing
constitutive relations, and geometric nonlinearity is at the stress-strain
level. Despite its advantages, such as being able to yield comprehensive
results and capture local responses, the finite element modeling approach is
typically cumbersome and computationally expensive for large-scale modeling
and post-processing. Member modeling approaches, i.e., macro-models, are
considered to be more convenient from this point of view, particularly for
practitioner engineers, due to their ease-of-use and computational efficiency.

Numerous macro-models are available in the literature that can make reasonably
accurate predictions for the structural response of reinforced concrete shear
walls. Macro-models are also referred to as discrete finite element models in
the literature [2]. Among macro-models, distributed and lumped plasticity
models are those that have been adopted by various seismic design standards
(e.g., ASCE 41-17 [3], Eurocode 8 [4]) and implemented into various analysis
software [5, 6, 7] for modeling of reinforced concrete shear walls. The
distributed plasticity model (also referred to as the fiber model) discretizes
wall cross-section using nonlinear uniaxial fiber elements, assuming that
plane sections remain plane after loading, whereas the lumped plasticity
approach adopts the use of elastic elements with effective stiffness and zero-
length plastic hinges at regions where material nonlinearity is expected to
occur (that is, typically member ends). When compared to the fiber model, the
lumped plasticity model has several disadvantages, such as neglecting (i)
deformations at section neutral axis during loading-unloading, (ii)
contribution of horizontal elements (e.g., beams, slabs) that are connected to
the walls, (iii) influence of the change in axial load on wall shear strength
and stiffness. Nevertheless, the method stands out with its ease-of-use and
computational efficiency in practice. It should also be noted that both models
have a common disadvantage, that is, shear-flexure interaction is not taken
into account as coupled axial–flexural (i.e., P-M) behavior is represented by
stress-strain relations of concrete and reinforcing steel along with generally
elastic or elastoplastic horizontal spring to reflect the shear behavior.
Other modeling techniques in literature account for shear-flexure interaction
based on strut and tie method [8, 9, 10, 11] and multiple vertical line
element model [10, 12, 13]; however, they are out of the scope of this paper.

Zero-length plastic hinges in the lumped plasticity model can be employed in
two forms: (i) rotational springs, (ii) fiber sections. This paper focuses on
the use of rotational springs. Nonlinear static and dynamic analyses are
conducted based on such plastic hinge models characterized by force-
deformation relationships, also known as "backbone curves", to ensure load and
deformation paths follow but not exceed them. Backbone curves are constructed
by connecting peak points at the first cycle of each increment [3], whereas
they can be approximated by a series of linear segments broken at critical
points for the sake of practicality (Fig. 1).

Figure 1: (a) Reversed cyclic test setup; (b) loading protocol; (c) load –
deformation response [14].

The history of the lumped plasticity approach dates back to the 1960s [15, 16,
17, 18], whereas the earliest introduction of the use of backbone curves in a
seismic performance-based evaluation guideline was in 1997 [19, 20]. In later
years, cyclic backbone curves for shear walls were recommended in FEMA 356
(2000) [21] and ASCE 41 (2006) [22], which updated the modeling parameters
based on the relevant research studies (Wallace et al. for squat walls [23]).
Definition of backbone curves and associated modeling parameters for
reinforced concrete shear walls in the most recent version of ASCE 41 [3] are
presented in Fig.2. The multilinear backbone curve model recommended for
shear-controlled walls includes five points at cracking (F), yielding (B),
peak strength (C), ultimate displacement (D), and residual strength (E). The
backbone curve model for flexure-controlled shear walls (Fig.2(b)) is less
refined (excludes the cracking point) as the deformations related to shear are
negligible compared to the deformations associated with flexure, whereas the
-x axis represents rotation (versus drift) for such walls. The normalized
shear force ($Q$/$Q$y) and drift ratio ($\Delta$/h) values at such points are
defined according to wall axial load level (as well as shear stress level for
flexure-controlled walls), where normalized shear force at cracking is
constant and equal to ($V$cr/ $V$y = 0.6) where $V$y equals to $V$n, i.e., the
nominal shear strength calculated based on ACI 318 [24]. It is worth noting
that the definition of backbone curves in ASCE 41 was very similar for
reinforced concrete columns up until ASCE 41-13 [25], where column modeling
parameters were used to obtain based on a combination of column axial load
level, reinforcement ratio, and shear stress ratio depending on the column
failure mode. However, a major revision has been made in ASCE 41-17, which now
provide parameters in the form of equations rather than the past table form.
This revision suggests that predictive equations ensure better accuracy and
easier use, whereas a similar study is needed for reinforced concrete shear
walls, in parallel with this study. Nonlinear modeling of reinforced concrete
shear walls based on lumped plasticity approach has been investigated by
various researchers for walls with different failure types. The lumped
plasticity approach was addressed in terms of plastic hinge length [26, 27,
28] and line-element models [29] for flexure-controlled walls, whereas semi-
empirical equations were developed for backbone curves of low-rise shear-
controlled walls by Carrillo and Alcocer [30]. More recently, Epackachi et al.
addressed the ASCE 41 cyclic backbone curve definitions by recommending
equations for peak shear strength for shear-controlled walls [31, 1].

As summarized above, traditional approaches are available in the literature to
define backbone curves of reinforced concrete shear walls; however, they
either require interpolation and generally provide conservative lower-bound
estimates with unknown dispersion (ASCE 41-17) or do not provide the entire
behavior characteristics (e.g., equations for lateral strength only [1]). Not
to mention, traditional approaches do not provide recommendations on modeling
parameters for walls with shear-flexure interaction (according to ASCE 41-17
[3], 1.5 $\leq$ $h$w/$h$w $\leq$ 3). This study aims to fill this gap by
approaching the problem from a machine-learning (ML) point of view, which has
not been explored before, and to propose predictive models to obtain cyclic
backbone curves of RC shear walls based on key wall design properties
(concrete compressive strength, horizontal boundary reinforcing ratio, etc).
The prediction of the backbone curves is essentially a multi-output regression
problem. Although the use of machine learning methods in the field of
structural analysis and material modeling date back to the late 1980s [32],
[33], such methods have received a lot of attention in the field of structural
earthquake engineering [34], in particular, for modeling the physical
relations of a structural system as an input-output problem via clustering,
classification, or regression [35], [36]. More recently, the machine learning
methods have been utilized to predict earthquake-induced building damage [37],
to define backbone curve models for reinforced concrete columns [38].

Figure 2: Generalized force-deformation relations based on ASCE 41-17 for a)
shear-controlled walls, b) flexure-controlled walls.

##  2 Experimental database

To develop predictive models for shear wall backbone curves based on the
machine learning approach, the key wall properties (Fig.3) and the critical
backbone variables are designated as input variables (features) and output
variables, respectively. To achieve this, more than 500 shear wall specimens
from various experimental studies conducted worldwide are collected based on a
review of available research [39, 40, 41]. Repaired or strengthened walls,
specimens tested under monotonic loading, as well as specimens constructed
with diagonal web reinforcement, confinement devices, composite materials
(e.g., I-section confinement), or hollow sections are excluded, resulted in
384 specimens useful for this study.

The database includes specimens that have different cross-sections
(rectangular [255/384] or barbell- shaped or flanged [129/384]), different end
conditions (cantilever [341/384] or rotations fixed on top [43/384]), and
different failure types (130 shear-controlled specimens, 126 specimens with
shear-flexure interaction, and 128 flexure-controlled specimens). Shear-
controlled specimens are those reported to fail in diagonal tension failure or
web crushing by reaching their shear strength before flexural yielding occurs,
whereas flexure-controlled specimens are those that yield in flexure before
reaching their shear strength, thus show damages such as concrete spalling and
crushing and/or reinforcing bar buckling at the boundary elements. Specimens
that contain both damage types are classified as walls controlled by shear-
flexure interaction. Fig. 3 presents the distribution of key design parameters
for each failure mode. Further details on the database can be obtained
elsewhere [42].

Figure 3: Distribution of key wall properties (384 specimens)

###  2.1 Definition of backbone curves

Backbone curve of each specimen is obtained using cyclic base shear-top
lateral displacement loops by taking the average of responses in positive and
negative regions, as shown in Fig. 4(a) for a representative specimen. The
deformation components are not separated; that is, the total deformation
included contributions from flexure, shear, and slip of the reinforcement.

Backbone curves include shear force and displacement at four critical stages,
namely: cracking, yielding, reaching shear strength, and reaching ultimate
displacement. The shear ($V$cr) and displacement ($\delta$cr) at the cracking
point are related through the elastic wall properties; therefore, the
following seven backbone variables are investigated in this study: shear at
cracking point ($V$cr), shear and displacement at yielding point ($V$y and
$\delta$y), shear and displacement at shear capacity ($V$max and $\delta$max),
as well as shear and displacement at displacement capacity ($V$u and
$\delta$u).

Figure 4: (a) Load – deformation response [14]; (b) backbone curve definition

Shear and flexure behaviors can be assessed together in Fig. 5(a) based on
backbone curves plotted with identifying colors for different failure modes.
Observations indicate that deformation capacity is influenced by the level of
shear strength as also shown in literature [39, 43, 44, 45]. Wall shear
strength degrades with increasing ductility, leading to shear failure (red
backbone curves), whereas, higher displacements can be reached under low shear
levels (blue backbone curves).

Fig. 5(b) demonstrates the distribution of the seven backbone variables in the
wall database. The seven backbone variables are compared to ASCE 41-17
recommended modeling parameters (a, b, c, d, e, f, g values shown in Fig.2) in
Tables 1 and 2 for shear-controlled and flexure-controlled walls,
respectively. Note that dominant wall types in Table 1 and Table 2 are
identified based on reported failure modes of the specimens, as data for walls
controlled by flexure based on ASCE 41-17 classification criteria (i.e.,
shear-controlled if $h$w/$l$w$<$1.5, flexure-controlled if $h$w/$l$w$>$3) is
too scarce to make an assessment. Comparisons indicate that ASCE 41 modeling
parameters achieve a fairly good agreement for walls subjected to low axial
load with the following exceptions for the sake of conservatism: (i) ASCE 41
neglects the significant increase in the slope after yield point (between $V$y
and $V$max), (ii) ASCE 41 residual strength recommendations ($V$u) are more
much lower for shear-controlled walls, (iii) ASCE 41 does not address the data
dispersion, therefore, can be too conservative in some cases. For other wall
types, higher discrepancies are observed except for well-predicted $f$, (i.e.
$V$cr/$V$y) value for shear-controlled walls. It is noted that ASCE 41 does
not provide modeling parameters for walls that have shear-flexure interaction.

Figure 5: (a) Backbone curves of 384 specimens in the database, (b) Distribution of seven backbone curve variables Table 1: Comparisons of database backbone values with ASCE 41-17 recommendations for shear-controlled walls Parameter | $\frac{(A_{s}-A_{s}^{\prime})f_{y}E+P}{t_{w}l_{w}f_{c}}$ | ASCE 41-17 | Database
---|---|---|---
$V$max/$V$y |  $\leq$ $0.5$ | $1.0$ |  $1.43$$\pm$$0.37$
| >0.5 | 1.0 | 1.18$\pm$0.05
$V$u/$V$max (c) |  $\leq$ 0.5 | 0.2 | 0.65$\pm$0.29
| >0.5 | 0.0 | 0.75$\pm$0.13
$V$cr/$V$y (f) |  $\leq$ 0.5 | 0.6 | 0.58$\pm$0.17
| >0.5 | 0.6 | 0.61$\pm$0.23
$\delta_{max}/h$ (d) |  $\leq$ 0.5 | 1.0 | 0.97$\pm$0.48
| >0.5 | 0.75 | 1.21$\pm$0.19
$\delta_{u}/h$ (e) |  $\leq$ 0.5 | 2.0 | 1.84$\pm$1.19
| >0.5 | 1.0 | 2.05$\pm$0.46
$\delta_{y}/h$ (g) |  $\leq$ 0.5 | 0.4 | 0.42$\pm$0.3
| >0.5 | 0.4 | 0.57$\pm$0.08
Table 2: Comparisons of database backbone values with ASCE 41-17 recommendations for flexure-controlled walls $\frac{(A_{s}-A_{s}^{\prime})f_{y}E+P}{t_{w}l_{w}f_{c}}$ | $\frac{V}{t_{w}l_{w}f^{\prime}_{c}E}$ |  | Confined
---
boundary
ASCE 41-17 | Database | ASCE 41-17 | Database
|  |  |  $V_{max}$/$V_{y}$ | c = $V_{u}$/$V_{max}$
<=0.1 | <=4 | Yes | 1.0 | 1.29 | 0.75 | 0.75
<=0.1 | >=6 | Yes | 1.0 | 1.38 | 0.4 | 0.70
>=0.25 | <=4 | Yes | 1.0 | 1.37 | 0.6 | 0.75
>=0.25 | >=6 | Yes | 1.0 | 1.22 | 0.3 | 0.73
<=0.1 | <=4 | No | 1.0 | 1.27 | 0.6 | 0.90
<=0.1 | >=6 | No | 1.0 | No data | 0.3 | No data
>=0.25 | <=4 | No | 1.0 | 1.34 | 0.25 | 0.62
>=0.25 | >=6 | No | 1.0 | No data | 0.2 | No data
$\frac{(A_{s}-A_{s}^{\prime})f_{y}E+P}{t_{w}l_{w}f_{c}}$ | $\frac{V}{t_{w}l_{w}f^{\prime}_{c}E}$ |  | Confined
---
boundary
ASCE 41-17 | Database | ASCE 41-17 | Database
|  |  | a = $\theta_{max}$-$\theta_{y}$ | b = $\theta_{u}$ -$\theta_{y}$
<=0.1 | <=4 | Yes | 0.015 | 0.011 | 0.020 | 0.019
<=0.1 | >=6 | Yes | 0.010 | 0.011 | 0.015 | 0.019
>=0.25 | <=4 | Yes | 0.009 | 0.008 | 0.012 | 0.016
>=0.25 | >=6 | Yes | 0.005 | 0.012 | 0.010 | 0.025
<=0.1 | <=4 | No | 0.008 | 0.013 | 0.015 | 0.020
<=0.1 | >=6 | No | 0.006 | No data | 0.010 | No data
>=0.25 | <=4 | No | 0.003 | 0.007 | 0.005 | 0.019
>=0.25 | >=6 | No | 0.002 | No data | 0.004 | No data

##  3 Background on Gaussian Process Regression

Gaussian Process Regression (GPR) is obtained by generalizing linear
regression. The GPR is a collection of random variables, with a joint Gaussian
distribution, hence is a probabilistic multivariate regression method. Let us
$D=\\{(x_{i},f_{i}),i=1,\ldots,n\\}$ refer to a given training set, where
$f_{i}=f(x_{i})$ is the value of the function at the sample $x_{i}$. The
Gaussian Process (GP), a generalization of the Gaussian probability
distribution, is represented as follows:

$f(\mathbf{x})\;\mathtt{\sim}\;GP(m(\mathbf{x_{i}}),k(\mathbf{x}_{i},\mathbf{x}_{j}))$
(1)

where $\mathbf{x}$ and $k(\mathbf{x}_{i},\mathbf{x}_{j})$ refer to mean and
covariance function, respectively. The covariance matrix express the
similarity between the pair of random variables and determined by means of
squared exponential (SE) kernel function given as

$k(\mathbf{x}_{i},\mathbf{x}_{j})={exp}(||\mathbf{x}_{i}-\mathbf{x}_{j}||^{2}/(2\sigma^{2}))$
(2)

where $\sigma$ is the kernel width that needs to be optimized. The predictive
function, $f_{*}i$ can be obtained for a given test sample $X*$, as following,

$\displaystyle
p(\mathbf{f}_{*}|\mathbf{X}_{*},\mathbf{X},\mathbf{f})=N(\mathbf{f}_{*}|\mathbf{\mu}_{*},\mathbf{\sigma}_{*})$
(3)

where

$\displaystyle\mu_{*}$ $\displaystyle=$
$\displaystyle\underline{k}_{*}^{T}.[K(X,X)+\sigma_{n}^{2}I]^{-1}\underline{y}$
(4) $\displaystyle\sigma_{*}^{2}$ $\displaystyle=$ $\displaystyle
k(\underline{X}_{*},\underline{X}_{*})-\underline{k}_{*}^{T}[K(X,X)+\sigma_{n}^{2}]^{-1}\underline{k}_{*}.$
(5)

$\underline{k}_{*}$ is a vector holding the covariance values between training
samples $X$ and the test samples $\underline{X}_{*}$. The $\sigma_{n}^{2}$ and
$\sigma_{x}^{2}$ refer to the noise variance and the confidence measure
associated with the model to output, respectively.

##  4 Implementation of Gaussian Process Regression and Results

###  4.1 Experimental Setup

The key wall properties (Fig.3) and the critical backbone variables are
designated as input variables (features) and output variables for the
regression analyses, respectively. Note that the problem is a multi-output
regression since multiple outputs are needed to be estimated.Training and test
datasets are randomly selected as disjoint sets from the database at a ratio
of 90% and 10%, respectively, and this random splitting is repeated a hundred
times. The purpose of a hundred trials is to assess the proposed method’s
generalization capability and robustness. Because of the randomization,
training/test pairings vary in each realization, resulting in a hundred
different learning models. The GPR-based models are constructed on each
training dataset, whereas the accuracy of the resulting models is evaluated
based on associated test dataset. The validation results are then averaged
over a hundred trials for each output. It should be noted that experiments
with fewer training datasets (e.g. 80% for training and 20% for testing) are
also considered; however, a large number of trials are overtrained, implying
that the amount of the training set is not sufficient to generate a reliable
learning model. Based on this experience, the size of the training set is
increased to 90% of all data.

The kernel parameters of the GPR method are automatically tuned using
gradient-based optimization routines to maximize the marginal likelihood.

The performance of models is evaluated based on three criteria: the mean
coefficient of determination (R2), relative root mean square error (RelRMSE),
and the prediction accuracy, that is, the ratio of predicted to actual values
(P/A). The experiments are conducted based on a hundred trials to ensure the
reliability and robustness of the proposed model, and the averaged results are
reported. All the codes are developed in Matlab [46] on a personal computer
with Dual Core Intel Core i7 CPU (3.50GHz), 16GB RAM, and MacOS operating
system.

Data pre-processing has been shown as a useful technique to improve the
quality of machine learning algorithms by transforming the data to a more
informative format [47]. Among several data transformation techniques
available, Box-cox and log transformation methods are applied to the outputs.
The original form of Box-Cox transformation equation [48] is as shown in Eq.
6.

$y(\lambda)=\left\\{\begin{array}[]{c}{\begin{array}[]{cc}{\frac{y^{\lambda}-1}{\lambda},}&{{\rm
if}\lambda\neq 0}\end{array}}\\\ {\begin{array}[]{cc}{\log(y),}&{{\rm
if}\lambda=0}\end{array}}\end{array}\right.$ (6)

where $y$ is the data value and $\lambda$ is a value between -5 and 5. The
optimal $\lambda$ is the one that transforms the data set to the best
approximation of normal distribution. The log transformation corresponds to
$\lambda$ equals zero. It is observed that the use of log transformation does
not improve the results, whereas Box-Cox (optimal $\lambda$ = -0.3) leads to
an increase in model accuracy for all outputs, particularly for $\delta$y,
$\delta$max, and $\delta$u (Table 3).

Table 3: Effect of data transformation on GPR model results (Averaged values over 100 trials). Output | No Transformation | Box-Cox Transformation
---|---|---
| R2 | RelRMSE | R2 | RelRMSE
$V_{y}$ | 0.80 | 0.36 | 0.84 | 0.33
$\delta_{y}$ | 0.72 | 0.55 | 0.82 | 0.50
$V_{max}$ | 0.87 | 0.28 | 0.89 | 0.28
$\delta_{max}$ | 0.84 | 0.43 | 0.91 | 0.33
$V_{u}$ | 0.57 | 0.64 | 0.62 | 0.59
$\delta_{u}$ | 0.77 | 0.40 | 0.89 | 0.29

###  4.2 Experimental Results and Discussion

Prediction accuracy and correlation values based on machine learning
experiments are summarized in Table 4 for each output. The results reveal that
the GPR method achieves better correlations (R2>0.8) for walls that failed in
shear (S) or shear-flexure interaction (SF), and the relatively low
correlation coefficients (mean R2 around 0.65) reduce the performance of
models. It is noted that modeling parameters in the horizontal axis of
backbone curves ($\delta$y, $\delta$max, $\delta$u) were also studied in terms
of chord rotation and curvature for flexure-controlled walls and drift ratio
for the shear-controlled walls; however, correlation coefficients were much
lower. Therefore, the results presented in Fig. 6 focus on walls that reached
their shear strength, that is, those controlled by shear or shear-flexure
interaction, and the prediction models are proposed for such walls for the
seven backbone variables.

Table 4: Averaged performances of the GPR-based models over 100 trials  Output |  |  | R2
---
(median)
| Mean

---
RelRMSE
| Mean
---
P/A
| Std. Dev.
---
P/A
$V_{cr}$ | S, SF | 0.71 | 0.44 | 1.08 | 0.47
All | 0.67 | 0.54 | 1.07 | 0.46
$V_{y}$ | S, SF | 0.84 | 0.33 | 1.02 | 0.29
All | 0.79 | 0.52 | 1.06 | 0.41
$\delta_{y}$ | S, SF | 0.82 | 0.5 | 1.06 | 0.46
All | 0.64 | 0.55 | 1.11 | 0.66
$V_{max}$ | S, SF | 0.89 | 0.28 | 1.04 | 0.26
All | 0.80 | 0.69 | 1.09 | 0.57
$\delta_{max}$ | S, SF | 0.91 | 0.33 | 1.05 | 0.38
All | 0.79 | 0.42 | 1.03 | 0.57
$V_{u}$ | S, SF | 0.62 | 0.59 | 1.3 | 0.99
All | 0.79 | 0.42 | 1.03 | 0.57
$\delta_{u}$ | S, SF | 0.89 | 0.29 | 1.03 | 0.29
All | 0.79 | 0.33 | 1.05 | 0.36
Figure 6: GPR model performances for the backbone variables over 100 trials.

Due to overtraining, trials with poor generalization capability such as those
with the highest correlation in training but very low correlation in testing,
are omitted, resulting in the mean values shown with the red diamond in Fig. 6
and the median values tabulated in Table 4.The results indicate that
coefficient of determination values are over 80$\%$ with low relative RMSE
values (around 0.3) for all seven backbone variables; except for the shear
force corresponding to ultimate deformation (i.e.$V$u), the mean R2 is around
60$\%$ with higher error. As seen in the boxplots, prediction accuracy is very
close to 1.0 with low standard deviations for all seven backbone variables,
except for the shear force at ultimate deformation. The lower performance for
$V$u implies that input-output relation can not be established for this
output, potentially because some specimens reached their ultimate deformation,
whereas, for the others, ultimate deformation is estimated at 80$\%$ of shear
strength, as mentioned earlier. This may have changed the structure of the
problem, resulting in relatively poor predictions.

Figure 7: Scatter plots for correlations of the proposed model for each
backbone variable

The correlation between the actual and predicted values is assessed based on
the proposed GPR-based predictive model that achieves the highest R2, the P/A
closest to 1.0, and the lowest error simultaneously (Fig. 7) for each backbone
variable. The scatter plots indicate that data are closely distributed around
the line of y = x, implying that the proposed models are able to make accurate
predictions when compared to the actual values. Results demonstrate that the
proposed GPR-based models achieve excellent performance in predicting shear
strength $V$max and corresponding displacement ($\delta$max) with highest
coefficients of determination (R2 equals to 0.96 and 0.97, respectively) and
smallest relative RMSE. The proposed model for ultimate deformation ($V$u)
also achieves good agreement with the actual data even though mean values over
100 trials were lower as this output is more data-dependent.

To evaluate the influence of each wall property (i.e. feature, Fig.3) on the
model output, optimized lengthscale hyperparameters of the kernel function are
used in the GPR model as each hyperparameter controls the spread of each
input. The hyperparameter optimization is carried out by computing the partial
derivatives of the negative log-marginal likelihood function with respect to
the given parameter. As small values show greater relevance, their inverses
are used for feature ranking. The most influential features are axial load
ratio (${P}/{({{A}_{g}}}f_{c}^{{}^{\prime}})$) and aspect ratio
(${h_{w}}/{l_{w}}$) for force-related backbone variables ($V$cr, $V$y, $V$max,
and $V$u), whereas shear span ratio (${M}/{Vl}_{w}$) is more critical for the
displacement variables ($\delta$y, $\delta$max, $\delta$u).

The performance of the proposed models is also compared to ASCE 41 modeling
recommendations based on 20 unseen test data in Table 5. Mean and standard
deviation of the predicted/actual (P/A) values are assessed as performance
indicators for each backbone variable, where ASCE 41 values are back-
calculated from the normalized modeling parameters. Mean P/A values are close
to 1.0 for all seven backbone variables for the GPR-based models, whereas
higher errors are observed based on ASCE 41. The standard deviation of the P/A
is also much smaller for the GPR-based models. Results indicate that the
proposed models outperform the traditional ASCE 41 approach for a wider
spectrum of walls (versus only low axial loads, as discussed in Table 1 with
higher accuracy.

Table 5: Performance of the proposed model for each backbone variable Output | ASCE 41 | GPR
---|---|---
| P/A mean | P/A dispersion | P/A mean | P/A dispersion
$V_{y}$ | 2.17 | 2.41 | 0.97 | 0.16
$\delta_{y}$ | 1.41 | 1.08 | 0.98 | 0.32
$V_{max}$ | 1.28 | 0.52 | 0.99 | 0.18
$\delta_{max}$ | 0.90 | 0.44 | 0.99 | 0.17
$V_{u}$ | 0.40 | 0.28 | 1.04 | 0.39
$\delta_{u}$ | 1.27 | 0.55 | 1.02 | 0.18

Machine learning experiments revealed that unusually thin ($l$w/$t$w$>$20)
shear walls and slender walls ($h$w/$l$w$>$3.0) decrease the correlation and
the prediction accuracy. The former is expected as typical $l$w/$t$w for shear
walls is reported as 13.3 [49] and reaches $l$w/$t$w=19 for around 350cm-long
shear walls [50] which corresponds to the maximum wall length in the database.
The latter factor decreasing correlations ($h$w/$l$w) can be associated with
the typical wall behavior, that is, squat wall responses are governed by shear
and slender wall responses are dominated by flexural behavior. However,
experimental results showed that squat walls may fail in flexure if adequate
reinforcement and detailing are provided [51], whereas slender walls may fail
in shear (web crushing) [52]. Therefore, backbone relations of slender walls
need to be analyzed separately even though they happened to fail in shear.

As noted above, the proposed GPR-based backbone curve models are generalized
and can be used regardless of failure modes. However, results demonstrate that
the predictions are more reliable especially for shear-controlled and shear-
flexure controlled walls. It is worth advising the potential users of the
proposed GPR-based models in that identification of dominant wall behavior
(failure mode) based on ASCE 41 may lead to conservative assessments. For
example, some moderate aspect ratio walls (1.5$<$ $h$w/$l$w$<$3) would be
classified as shear-flexure controlled even though they would actually fail in
flexure, creating a false impression that backbones of such walls can be
predicted based on the proposed GPR-based models. Therefore, prior to the use
of the proposed models for a new shear wall specimen, a more accurate failure
mode prediction [53, 54] is recommended.

It is also noted that other machine-learning methods including k-nearest
neighbour regression (KNN) [55], least absolute shrinkage and selection
operator (LASSO) [56], neighbourhood component analysis (NCA) [57], random
forest [58], regularized logistic regression (RLR) [59], and support vector
regression (SVR) [60] were also utilized in this study in an attempt to obtain
higher correlations and accuracies; however, none of them were as accurate as
the GPR method.

##  5 Summary and Conclusions

Backbone curves are critical component modeling tools in element-based
nonlinear modeling to represent the response under reversal drift cycles. It
is essential to simulate the hysteretic response as accurately as possible to
achieve realistic assessments as nonlinear static and dynamic analyses are
conducted based on such curves.

This study aims to address this need by developing predictive models for
backbone curves based on a comprehensive shear wall experimental database
using a powerful machine learning method, that is Gaussian Process Regression
(GPR). Seven backbone curve variables associated with four limit states,
namely: cracking, yielding, peak shear strength, and ultimate deformation
capacity are predicted in terms of twelve wall design properties (e.g.
concrete compressive strength, reinforcement details, wall geometry). To
achieve this, the backbone curve variables and wall design properties are
designated as outputs and inputs, respectively. The database is divided into
train and test data sets, whereas the performances of the proposed models are
evaluated based on statistical metrics (e.g. coefficient of determination,
relative root mean square error, prediction accuracy) based on test data sets.
The prediction performances of the proposed GPR models are also compared to
traditional modeling approaches (ASCE 41-17) and shown to be superior.

For best performance, the proposed models are recommended for walls that
satisfy the following conditions: (i) wall behavior is controlled by shear or
shear-flexure interaction, (ii) $l$w/$t$w$\leq$20, and (iii) $h$w/$l$w$\leq$3.

The results reveal that one can predict the shear force - displacement
relations, thus, the existing capacity and seismic performance of RC shear
walls, close to accurate using the proposed GPR-based models. This study will
contribute to the literature by providing tools that achieve dual objectives
of high accuracy performance and computational efficiency to ensure more
realistic performance evaluations.

##  6 Acknowledgements

The project has been supported by funds from the Scientific and Technological
Research Council of Turkey (TUBITAK) under Project No: 218M535. Opinions,
findings, and conclusions in this paper are those of the authors and do not
necessarily represent those of the funding agency.

## Notation

${{A}_{g}}$ | = gross cross section area of wall
---|---
${\delta}_{cr}$ | = cracking displacement
${\delta}_{y}$ | = yield displacement
${\delta}_{max}$ | = displacement when peak shear is reached
${\delta}_{u}$ | = ultimate displacement
$\varepsilon$ | = the noise term
$E\left(X|Y\right)$ | = expected value of random variable X given Y
$f_{c}^{{}^{\prime}}$ | = concrete compressive strength
${f}_{ybl}$ | = longitudinal boundary reinforcement yield strength
${f}_{yl}$ | = longitudinal web reinforcement yield strength
${f}_{ysh}$ | = transverse boundary reinforcement yield strength
${f}_{yt}$ | = transverse web reinforcement yield strength
${h}_{w}$ | = wall height
${h_{w}/l_{w}}$ | = aspect ratio
$\mathbf{I}$ | = the identity matrix
${l}_{w}$ | = wall length
$\mathcal{L}$ | = the loss function
$m$ | = number of specimens
${M}/{Vl}_{w}$ | = shear-span ratio
$n$ | = number of design parameters
${P}/{({{A}_{g}}}f_{c}^{{}^{\prime}})$ | = axial load ratio
$\mathbf{P}\left(A|B\right)$ | = probability of event A given event B occured
${s}$ | = stirrups spacing
${t}_{w}$ | = wall thickness
${{\rho}_{bl}}$ | = longitudinal boundary reinforcement ratio
${{\rho}_{l}}$ | = longitudinal web reinforcement ratio
${{\rho}_{sh}}$ | = transverse boundary reinforcement ratio
${{\rho}_{t}}$ | = transverse web reinforcement ratio
$\mathbf{w}$ | = weighing vector of features
$\mathbf{X}$ | = $m\times n$ dimensional matrix holding all the features
${{x}_{i}}$ | = $i^{th}$ feature value of the ML model
$\mathbf{y}$ | = $m$ dimensional output vector
${V}_{cr}$ | = shear force at cracking
${V}_{y}$ | = shear force at yielding
${V}_{max}$ | = peak shear force
${V}_{u}$ | = shear force when ultimate displacement is reached

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[a,b]Diego Blas

#  Quantum technologies for fundamental (HE) physics

###### Abstract

In this brief contribution I will highlight some directions where the
developments in the frontier of (quantum) metrology may be key for fundamental
high energy physics (HEP). I will focus on the detection of dark matter and
gravitational waves, and introduce ideas from atomic clocks and magnetometers,
large atomic interferometers and detection of small fields in electromagnetic
cavities. Far from being comprehensive, this contribution is an invitation to
everyone in the HEP and quantum technologies communities to explore this
fascinating topic.

AION-REPORT/2023-12

##  1 Introduction

The field of quantum technologies for fundamental physics is going through a
revolution. The reasons are multiple. First, the progress at all levels in the
realm of quantum technologies motivates an intense exploration of how the high
precision and control achieved can be used to detect the subtle traces of new
particles or other new phenomena. Also, different more classical approaches to
the detection of dark matter or gravitational waves are exhibiting their
limits, and quantum technologies offer new exciting possibilities to extend
them. Examples of this excitement are: the CERN Quantum Technology Initiative
(CERN QTI, https://quantum.cern/), the Physics Beyond Colliders Study Group
(https://pbc.web.cern.ch/), the Task Force 5 (Quantum and Emerging
Technologies) of the ECFA Detector R&D Roadmap Symposium
(https://indico.cern.ch/event/999818/), the Fermilab Quantum Institute
(https://quantum.fnal.gov), the UKRI Quantum Technologies for Fundamental
Physics (https://uknqt.ukri.org/our-programme/qtfp/), the Fundamental Physics
Program Analysis Group (FunPAG) of NASA (https://www.jpl.nasa.gov/go/funpag)
or the reviews [1, 2, 3]111If the reader misses other interesting
initiatives/references, please contact me to add them..

In this contribution I shall focus on the metrology and/or quantum sensing
part of this effort when applied to the search of new fundamental backgrounds.
The first great advantage of this approach as compared to other detectors
(e.g. those that need some minimal momentum transferred to generate a signal)
is that quantum mechanically any scattering event leaves an imprint in the
final states (e.g. in their phases), even when the momentum transferred
vanishes [4]. In this sense, large fluxes of low-energy particles are an ideal
target for these technologies. Furthermore, some of the intrinsic quantum
phenomena (such as entanglement) are fragile, and subject to work as sensors
of any interacting background. The combination of this fact with the possible
presence of coherent effects may also produce very sensitive detectors.
Finally, quantum noise is already playing a role in several measurements, e.g.
in ground-base interferometers [5]. The ability to manipulate our sensors may
beat this quantum noise in some situations, e.g. [6], and understanding the
limitations of how much the quantum noise can be reduced is essential for
future detectors.

Another important aspect of the program is that its boundaries are far from
fixed. The fast development of our ability to understand and manipulate
quantum phenomena is spurring the imagination of several groups that try to
leverage it towards new detection possibilities. As an example of this
development, one can consider atomic clocks: the fractional frequency
uncertainty has been consistently improving since 1960s in the case of
microwave clocks, and very dramatically since 1990s for optical clocks [7].
When considering the growing investment into the field (compare e.g. this
study222Another source of information of the global effort in quantum
technologies is https://thequantuminsider.com/2023/02/17/quantum-
technology-2022-investment-update-key-trends-and-players/. for 2023:
https://qureca.com/overview-of-quantum-initiatives-worldwide-2023/ with the
similar one from 2020: https://qureca.com/es/the-quantum-ecosystem-and-its-
future-workforce-2/), the possibilities for the future seem extraordinary.

Before moving to the main body of this contribution I want to make a couple of
remarks. First, I want to acknowledge that quantum technologies have been used
in HEP detectors for many years. The main difference of more recent effort is
that the latter put the _quantum technology at the center of the detection
strategy_ , which typically also means focusing on table-top set-ups (at least
as a starting point). Second, a sub-field which I find very promising, but
that I will touch only very briefly, is the use of quantum sensors to detect
“machine-made” backgrounds (e.g. by placing them close to reactors or beam-
dumped experiments). The small size of these devices and their sensitivity to
small momentum transfer may be key in searches for new physics in these set-
ups.

##  2 Fundamental cosmological backgrounds

Our Universe has been rather generous in the fundamental cosmological (and
astrophysical) backgrounds that arrive to Earth. One of the best known
examples is the cosmic microwave background (CMB) generated when recombination
happened and the Universe became transparent [8]. This signal (today detected
in the microwave band) carries very precise information from 13.7$\times
10^{9}$ years ago, but also from the intervening structure that it found as
travelling towards us.

How do we access more primordial information or from places that have obscured
the CMB signal or that do not produce detectable photons? A possible answer is
by using _neutrinos_ , that can have very long mean-free paths and for which
the Universe was almost transparent at recombination. The equivalent of the
CMB in this case is the cosmic neutrino background (CNB): as the Universe’s
temperature dropped below few MeV (which happened much earlier than
recombination), the neutrinos that participated in nuclear reactions saw an
almost transparent universe, and propagate from this moment almost unimpeded
[9]. The searches for the CNB have so far proved fruitless, in part due to the
small typical momentum of its constituents, which means a tiny available
energy to generate recoils and very small cross-section of the CNB with our
detectors. One of the most promising directions is PTOLEMY [10], though it is
not yet clear when the CNB will be found (see for instance [11, 12, 13]). The
quantum frontier may certainly be explored to this aim and to answer other
questions about neutrinos: their mass, their nature (Dirac or Majorana), more
insights on their family structure or possible new interactions with standard
model fields (photons, in particular). For this direction, one could not only
use cosmological neutrinos, but also machine-made fluxes. As described above,
I will not discuss this fascinating possibility in any detail in this
contribution.

A more uncertain, yet existing, background is that of gravitational waves
(GWs). GWs are in principle produced by every energetic process in the
Universe. This universality, and the fact that they travel without basically
interacting through cosmological scales, makes of them ideal messengers of
some of the most extraordinary events taking place in the cosmos. For
instance, they are copiously generated in the dynamics of many scenarios of
the primordial Universe (inflation in particular) [14] or by the merger of
black holes at more recent times, that do not leave any other observable
signal. Their first direct detection took place in 2015, and it corresponded
to GWs in a window close to 100 Hz and emitted by a merger of black holes
[15]. Since then, many other detections have followed in the 10 Hz to 10 kHz
band. In 2023, a new signal was confirmed at nHz [16, 17, 18, 19]. There is a
global effort to detect GWs at other frequencies, with CMB studies [20] and
the ESA lead mission LISA [21] as spearheads. Quantum technologies may impact
this search in several ways. Besides increasing the sensitivity of current
detectors, they can play a pivotal role to access _higher frequencies_ , which
may have a direct impact in laboratory set-ups. I will discuss this further in
the next section. A better coverage of the spectrum of GWs will have a very
strong impact in our understanding of Nature, which motivates the widest
possible search.

The final possible fundamental background I will discuss is that coming from
the dark sector. Our current understanding of the Universe requires the
existence of a dark matter (DM) component, five times more abundant than
ordinary matter, and a dark energy component controlling the current
accelerated expansion observed at cosmological distances [22]. Very little
more is known about them, besides their expected energy density at different
places, including the Solar system [23]. Properties such as the mass, spin,
interactions with standard model fields, state (classical field, particle,
compact object…)… of the particles that may be responsible of these ‘dark’
sectors are rather unconstrained. Coming back to the logic of Sec. 1, the
possibility to control very fragile systems opens a battery of new techniques
to directly access the dark sector. In the next section I will only discuss
the influence of some DM models in selected set-ups. It would be very useful
to extend this to all possible models (of DM, but also of dark energy) and
set-ups.

##  3 Deriving the influence of fundamental backgrounds in precision
measurements

When one wants to take the next step in the program that I am describing, a
_model agnostic_ approach is possible, based on ideas from _effective field
theory_. The idea can be summarized in the following steps: _i)_ One decides
the fundamental properties of the field of interest (e.g. spin or mass). _ii)_
One decides the symmetries that govern the interactions. _iii)_ One writes
_all_ possible interactions of these new fields with ordinary matter, and
classifies them according to their relevance. Here want to understand the
coupling to the degrees of freedom in laboratory set-ups, which are normally
related to atomic physics and light. _iv)_ One derives the limit of interest
for the laboratory set-ups. Typically, this means the non-relativistic
approximation for matter. _v)_ One studies the consequences of these operators
(“interaction Hamiltonians”) one by one, and for the different possible states
of the background (considered e.g. as a classical field, collection of
particles, etc.). Possible effects may be those of new absorption channels,
scattering processes, decays, etc. _vi)_ One finally connects the conclusions
to the possible models including them.

I believe that this program can be addressed in a rather generic way, and
generate a useful dictionary to catalyze the communication between the HEP and
the quantum sensing communities. I will now use the interaction of light with
matter as an example of how this could work. Let us imagine that we want to
detect photons without knowing their properties, beyond the fact that they are
spin-1 particles (not even their mass). One can postulate that they couple to
electrons through an associated operator. The set of allowed operators in
$A^{\mu}$, in increasing dimension, starts as

$q_{1i}A^{\mu}\bar{\psi}_{e}O^{i}_{\mu}\psi_{e},~{}~{}~{}~{}~{}q_{2i}\frac{\partial^{\nu}A^{\mu}}{\Lambda}\bar{\psi}_{e}O^{i}_{\mu\nu}\psi_{e},~{}~{}~{}~{}q_{3i}\frac{A^{\nu}A^{\mu}}{\Lambda}\bar{\psi}_{e}\tilde{O}^{i}_{\mu\nu}\psi_{e},...$
(1)

where $O^{i}_{\mu}$, $O^{i}_{\mu\nu}$ and $\tilde{O}^{i}_{\mu\nu}$ are
operators acting on the spinor $\psi_{e}$ and associated to a unitary theory.
$\Lambda$ is a scale that controls the relevance of the second and third
operators with respect to the first one (we have assumed that all $q_{ni}$
have the same dimensionality, as do $O_{\mu}$, $O_{\mu\nu}$ and
$\tilde{O}^{i}_{\mu\nu}$). These coefficients are a priori arbitrary and their
effects may be studied independently. If one focuses on electrodynamics, then
it is the term with $O^{i}_{\mu}=\gamma_{\mu}$ which is relevant. This analogy
is very important, because some of the effects we may be looking for have
already been considered as possible backgrounds or foregrounds for the
devices. For instance, in [24] the effect of a background gas collisions is
studied for atomic clocks. The last operator in (1) may be constrained with
this observation. For quark couplings, one needs to connect them to nuclei
through some standard (quite complex) techniques, that are eventually
summarize in a collection of form factors, e.g. [25].

The forth step would be to reduce these operators to those relevant in the
field of interest. For instance, in atomic physics our electrons will be (most
of the times) non-relativistic, so we can simply evaluate the biliniars of the
form $\bar{\psi}_{e}O^{i}_{\mu}\psi_{e}$ for arbitrary spinors, and take the
non-relativistic limit, e.g. [26]. The final result will be a set of non-
relativistic operators, depending on the mass($m$), momentum ($\vec{p}$),
position ($\vec{x}$) or spin ($\vec{S}$) of the electron (further
contributions may appear for non-fundamental particles),

$\tilde{q}_{1}A^{i}p^{i}_{e},~{}~{}~{}~{}~{}\tilde{q}_{2}\frac{\epsilon_{ijk}\partial_{j}A_{k}}{\Lambda}S^{i}_{e},~{}~{}~{}~{}\tilde{q}_{3}\frac{A_{i}A_{j}}{\Lambda}x^{i}_{e}x^{j}_{e},...$
(2)

For the fifth step, one also needs to decide the state of the new field
($A^{\mu}$ in the previous example). It can be a massive or massless field,
which may imply that it may also be described in a non-relativistic regime. In
this case, one can proceed as for the fields of the detector. More relevant
are the properties as a function of time, and in terms of the occupation
number of its constituents. For instance, the field $A^{\mu}$ may be in a
configuration with large occupation numbers and random phases per phase space
element, and a classical description may be convenient. The new field
$A^{\mu}$ may also interact as _individual particles_ , and treated as a
quantum field. In this case, each of these particles may go through a region
of the device, which one can quantify by understanding the size of the
corresponding wave-packet, and the average effect of the flux of incoming
particles. This requires to study in detail how the whole wave function of the
elements of the device evolve with time. An example that may be illuminating
is the work for atomic clocks and dark matter particles in [27, 28]. With
these operators, we can proceed and study their consequences (with our quantum
technologies colleagues). The final step reinterprets these effects in terms
of the original proposals.

For this fifth step, one needs to use several properties of the backgrounds.
For, instance for dark matter we expect an energy density of $\rho\approx
0.3\,$GeV/cm3 and an average velocity of $\bar{v}\approx 10^{-3}$ due to the
motion of the Sun in an otherwise basically isotropic background of DM, which
can produce effects of the from $\alpha S_{e}^{i}mv_{\rm DM}^{i}$ If this is
the case, the fact that the average velocity does not cancel, will generate a
non-vanishing coupling that may be confused with an anomalous magnetic field.
However, there are handles to break this degeneracy: the coupling to different
species of standard fields, or the fact that the effect will be modulated
yearly and daily as we travel through the DM galactic halo. Note also that
even in the case of a non-coherent total effect, this term will act as a noise
source for the determination of spin measurements, eventually acting as a
source of uncertainty that one can try to search for.

The previous steps can be formalized and organized to fit a single review
where all (or almost all) relevant effects may be summarized. This sort of
_dictionary_ would be very useful for a fruitful dialogue with the community
of quantum technologies and I hope that we are able to do this in the upcoming
years.

##  4 Three (biased) examples

In this section, I will briefly comment on three examples where the new
frontiers in (quantum) metrology are playing a relevant role for the program I
described above.

###  4.1 Dark matter and cosmic neutrinos with atomic clocks and co-
magnetometers

It is well-known that light dark matter may not have enough momentum to
generate recoil events in traditional dark matter detectors. More concretely,
for dark matter scattering with a nucleus, the recoil energy is at most
$E_{R}^{\max}\sim\left(\frac{m_{\chi}}{\mathrm{GeV}}\right)\mathrm{keV}$,
which may be below threshold to be observed in traditional searches [29]. The
quest to observe light dark matter requires new ideas [30]. The reason why
quantum detectors may yield a new handle in this problem is that _even at
vanishing momentum transfer_ the scattering of particles has an impact on the
phase of the scattering process. Hence, there is no real threshold in these
experiments, as long as we are able to detect the effect of the scattering in
the phase of the scattered particles333This is not exactly true, as the phase
can not be determined with arbitrary precision in a system with a finite
number of degrees of freedom [31]. See e.g. [32] for a recent discussion of
quantum measurements in fundamental physics.. This was studied in detail in
[27, 28] (see also [33]). In these works, we studied how the transition
probability is modified in an interferometry process by the scattering of DM
particles of very light mass. The latter is the basis for atomic clocks and
atomic magnetometers, and our results show how these set-ups may yield
spectacular constraints for many models of particle DM444Atomic clocks and
magnetometers have been proposed to look for ultra-light dark matter, where
the DM component behaves as a classical field [34, 35]. We still work in the
particle regime, and it is the collection of many scattering events which
generates the signal.. I will only briefly mention the case of co-
magnetometers, and refer the reader to the previous references for more
details. Atomic magnetometers are based on a sample of $N\sim 10^{21}$
polarized atoms, whose spin interacts with an external magnetic field through
an operator $H_{\rm int}=-\gamma\vec{B}\cdot\vec{\lambda},$where $\lambda$
represents the spin of the atom, and $\gamma$ is the effective coupling. As a
result, for a polarized sample where $\vec{\lambda}$ is not aligned with
$\vec{B}$, $\vec{\lambda}$ of the atoms will precess with a frequency
$\omega\equiv\gamma B$, which is connected to the difference in the energy
levels generated by (4.1). If a flux of DM goes through the sample and
scatters through it, the forward scattering limit can be interpreted as a
modification of the effect of the free Hamiltonian, and one can show that the
effective Larmor frequency is now

$\omega_{\rm DM}\equiv\gamma\beta=\gamma\left(B+\frac{2\pi
n_{\chi}}{m_{\chi}\gamma}\left(\bar{f}(0)_{1}-\bar{f}(0)_{2}\right)\right),$
(3)

where $f(0)_{i}$ is the matrix element corresponding to forward scattering of
the different spin states with $i$. Here $n_{\chi}$ is the DM number density
and $m_{\chi}$ the DM mass and the bar indicates the average over the
scattering events during the experiment. This last concept is quite different
as compared to other searches: since we are sensitive to the changes in the
phase, it is important that the scattering events do not average to 0555As
mentioned above, a certain level of induced “noise” is always expected.
Furthermore, other effects, such as the lost of coherence, are directly
affected by the total number of scattering events. , which may happen if the
interaction depends on arbitrary parameters such as the DM spin. As far as I
know, this approach has not been fully exploited to search for DM, and I
invite the reader to find more details in [27, 28, 33]. Similarly, it seems a
good possibility to find new ways to explore the CNB. Our results of [28]
implied that we could win several orders of magnitude if the CNB is polarized
or has some non-trivial chemical potential. Even in these non-standard cases,
we are very far from claiming that this is a promising direction.

Before moving to the next example, I want to mention that these ideas could
also be used to detect large backgrounds/fluxes which may be produced in
reactors or accelerators (e.g. at beam-dumped configurations, such as those
described in [36]), and go through the quantum devices without the need to
deposit their energy. I am not aware of any publication in this direction.

###  4.2 Large atomic interferometers

The use of large atomic interferometers (AI) to detect dark matter or
gravitational waves has received growing attention in the last years, e.g.
[37, 38] (see also [40, 41, 39, 42, 44]). The idea of these techniques can be
summarized as follows: a sample of atoms, which can be in states $|e\rangle$
and $|g\rangle$ with energy difference $\omega$, are launched in the
$|g\rangle$ state and evolve in free fall, except at some times when a laser
generates $\pi/2$ or $\pi$ pulses. The first of these pulses (at $t=0$) split
the beam into $|g\rangle$ and $|e\rangle$ atoms and imprint a momentum kink to
these last ones. The $\pi$ pulse (at $t=T$), when performed carefully, turns
the $|e\rangle$ portion of the sample into $|g\rangle$, and viceversa, while
generating a negative momentum kick to the $|e\rangle$ states, and positive to
the $|g\rangle$ states. Finally, another $\pi/2$ pulse is performed at $t=2T$
(see [37, 38, 40, 41, 39, 42, 44] for more details). The final result of all
these processes, if unperturbed, generates a wavefunction with different
phases for the two samples of atoms

$\left.|\psi\rangle\sim\frac{1}{\sqrt{2}}\left(\left(1+e^{-i\Phi_{\mathrm{TOT}}}\right)|g\rangle+\left(1-e^{-i\Phi_{\mathrm{TOT}}}\right)|e\rangle\right)\right).$
(4)

In the previous expression $\Phi_{\mathrm{TOT}}\sim\omega T$, where $T$ is
defined above. As a result, the frequency $\omega$ is imprinted in the final
state, and by measuring the population in each state, we can be sensitive to
the energy difference and $T$. When we consider that the previous processes
happen in the presence of a DM background, the same physics considered in the
previous section does in principle modify the energy split between the two
states if the DM particle distinguishes between them as it scatters. For
instance, (4.1) would generate an extra shift in the energy difference of the
states of different spin. Other DM candidates may simply change the
fundamental constants, and hence the energy levels. For instance, a coupling
$\phi m_{e}\bar{\psi}_{e}\psi_{e}$ (quite standard for ultra-light dark matter
[38]) would change the mass of the electron if the field $\phi$ has some time
or space-dependent properties (e.g. if it behaves as a non-trivial classical
field). As should be apparent from the the previous section, the scattering of
DM of higher masses can also generate interesting phenomena in AI [33]. I
believe that there is still plenty of exploration space in this direction. For
instance, AIs are also fantastic accelerometers, and this may yield a new
handle to discover light dark matter [45].

Another very interesting use of AIs is as explorers of GWs. This is simple to
understand from the fact that the the light travel time $L$ in will be
modified in the presence of GWs, impacting the previous processes. The
relevance of this possibility is enhanced by the fact that the set-ups that
have been suggested for the following years will be sensitive to GWs in a band
very hard to cover by other methods (a bit below Hz, the mid-band between
space-based and ground-based detectors) [37, 46].

To the interested reader, here is a (probably incomplete) selection of
proposals: the MAGIS-100 detector [47] (currently developing a 100-m vertical
set-up); the MIGA project [48] (currently developing a horizontal set-ups of
$\sim 200$m); the ELGAR initiative [42]; the ZAIGA proposal [49] (aiming at a
300-m vertical set-up); the AION collaboration [44] (currently building the
10-m vertical set-up); and the futuristic AEDGE [50], proposing atomic
interferometry in space. A lot of new collaborations are possible, in
particular regarding where to install the future set-ups [38, 51].

###  4.3  GWs in (superconducting radio-frequency) cavities

I will conclude with a brief discussion of another idea where cutting-edge
(quantum) metrology may play a very important role for fundamental physics. As
I discussed in Sec. 2, the study of the widest possible spectrum of GWs is a
very important challenge of modern physics. By accessing frequencies above 10
kHz (those scrutinized with ground-based detectors – the LIGO-Virgo-KAGRA
collabororation), one explores a plethora of dark matter and early universe
models very hard to probe otherwise [52]. Since signals in the range MHz to
GHz or even higher are conceivable, this implies a direct connection with the
time and length scales of laboratory set-ups, and hence an exciting
possibility of testing high frequency gravitational waves (HFGWs) with quantum
sensors.

Form the equivalence principle, we know that any source of energy and momentum
couples to gravitation. As a result, an abundance of effects can be expected
to be induced by the passage of a HFGW in a very precise device. I will only
comment on one of these ideas: the effect of a HFGW as it impacts an
electromagnetic cavity in the presence of background electromagnetic (EM)
fields $A_{b}^{\nu}$. Two main processes may occur: the first one is that as
the GW interacts with the background EM field, it generates an effective
current $j_{\mu}$ linear in $h^{\alpha\beta}$ and $A_{b}^{\nu})$ that sources
a new “signal” EM field. This operator $A^{\mu}_{s}j_{\mu}$ is easy to
understand as the energy and momentum that GWs couple to are quadratic in
$A^{\mu}$. Quite relevantly, this coupling is similar to the one natural for
axionic dark matter ($a$), $a\,\vec{E}\cdot\vec{B}$ [53], which means that a
whole battery of techniques searching for these fields is ready to be used to
search for GWs. Furthermore, the GW shakes the walls of the cavity666GWs of
amplitude $h$ change “physical” distances as $\delta L\sim hL$., changing its
geometry and affecting the properties of the modes that may be stored in
$A_{b}^{\nu}$ (which may generate mode-mixing). Due to lack of space (time in
my talk), I cannot describe these set-ups in more detail. I would still convey
the message that current studies show a vast parameter space that may be
explored in the near future, where new surprises from Nature may be waiting
[54, 55, 56, 52].

##  5 Summary and future directions

Quantum sensing/devices provide new ways to detect foregrounds/backgrounds of
relevance for fundamental physics (e.g. gravitational waves, neutrinos or dark
matter and dark energy). A key characteristic of why this can generate
breakthroughs in the near future are the low energy/momentum transferred
thresholds for quantum effects to happen, which are ideal to sense
“substantial” fluxes with tiny cross-sections. I believe that as HEP
practitioners, we have a very important task in front of us to transform our
models of interest into predictions for the degrees of freedom that (quantum)
sensing practitioners recognize. In particular, by generating a dictionary
from HEP models into interacting Hamiltonians $H=H_{0}+H_{\rm int}^{\rm new\
physics}$ we may bridge an important gap and generate a fluid dialogue between
communities.

In this short (and inevitably biased) contribution, I presented some examples
where we started to fill this gap for searches of dark matter and GWs in co-
magnetometers, atomic clocks, long-baseline interferometers and EM cavities
(with a couple of comment on beam-dumped experiments and neutrino searches). I
am convinced that we are just starting to grasp the potential of the quantum
world for fundamental physics, and the use of ideas such as faster readouts,
single-photon detectors, larger or entangled samples, will take us towards
totally unexpected consequences. I hope that this contribution is intriguing
and interesting enough to motivate future directions towards new milestones in
our understanding of Nature.

Acknowledgements. It is a pleasure to thank L. Badurina, S. Ellis and C.
Murgui for comments on a first draft of this contribution. I am supported by a
‘Ayuda Beatriz Galindo Senior’ from the Spanish ‘Ministerio de Universidades’,
grant BG20/00228. The research leading to these results has received funding
from the Spanish Ministry of Science and Innovation
(PID2020–115845GB-I00/AEI/10.13039/501100011033). IFAE is partially funded by
the CERCA program of the Generalitat de Catalunya. I also acknowledge the
support from the Departament de Recerca i Universitats de la Generalitat de
Catalunya al Grup de Recerca i Universitats from Generalitat de Catalunya to
the Grup de Recerca 00649 (Codi: 2021 SGR 00649).

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