# Accurate battery lifetime prediction across diverse aging conditions with
deep learning

Han Zhang1⋆, Yuqi Li2⋆, Shun Zheng3, Ziheng Lu3, Xiaofan Gui3, Wei Xu1, Jiang
Bian3 1Institute for Interdisciplinary Information Sciences, Tsinghua
University2Department of Materials Science and Engineering, Stanford
University3Microsoft Research

(2018)

###### Abstract.

Accurately predicting the lifetime of battery cells in early cycles holds
tremendous value for battery research and development as well as numerous
downstream applications (NE, ; attia2021statistical, ; CLO, ; Yao2023NRM, ).
This task is rather challenging because diverse conditions, such as electrode
materials, operating conditions, and working environments, collectively
determine complex capacity-degradation behaviors. However, current prediction
methods are developed and validated under limited aging conditions (NE, ;
attia2021statistical, ; HUST, ), resulting in questionable adaptability to
varied aging conditions and an inability to fully benefit from historical data
collected under different conditions. Here we introduce a universal deep
learning approach that is capable of accommodating various aging conditions
and facilitating effective learning under low-resource conditions by
leveraging data from rich conditions. Our key finding is that incorporating
inter-cell feature differences, rather than solely considering single-cell
characteristics, significantly increases the accuracy of battery lifetime
prediction and its cross-condition robustness. Accordingly, we develop a
holistic learning framework accommodating both single-cell and inter-cell
modeling. A comprehensive benchmark is built for evaluation, encompassing 401
battery cells utilizing 5 prevalent electrode materials across 168 cycling
conditions. We demonstrate remarkable capabilities in learning across diverse
aging conditions, exclusively achieving 10% prediction error using the first
100 cycles, and in facilitating low-resource learning, almost halving the
error of single-cell modeling in many cases. More broadly, by breaking the
learning boundaries among different aging conditions, our approach could
significantly accelerate the development and optimization of lithium-ion
batteries.

††copyright: acmcopyright††journalyear: 2018††doi:
XXXXXXX.XXXXXXX††conference: ; ; ††⋆ Equal contribution. This work was done
during the internships of Han Zhang and Yuqi Li at Microsoft Research.

Owing to their high energy densities and low production costs, lithium-ion
batteries have been widely adopted in modern industry, propelling the surge of
renewable energy solutions and electric vehicles(Tarascon2001IssuesAC, ;
Armand2008BuildingBB, ; Dunn2011ElectricalES, ). Nevertheless, the capacity of
lithium-ion batteries inevitably fades with cyclic operations due to their
intrinsic electrochemical mechanisms. Unexpected rapid degradation not only
leads to poor user experiences, such as range anxiety for electric vehicles,
but can also affect the operation of essential facilities, such as the
stability of power grids. To proactively mitigate these side effects,
accurately predicting battery lifetime in early cycles has been identified as
a critical task(ng2020soc_soh_nmi, ; Yao2023NRM, ). This task is rather
challenging because numerous factors, including electrode materials, charging
and discharging protocols, and working environments, collectively influence
the complex battery aging process. Recent data-driven approaches that leverage
machine learning have made remarkable progress in this direction (NE, ;
attia2021statistical, ; CLO, ; HUST, ), identifying critical electrical
features that highly correlate with cycle life.

However, existing methods for battery lifetime prediction have been developed
and validated under limited aging conditions, such as testing only lithium-
iron-phosphate ($LiFePO_{4}$) batteries and using single charging or
discharging protocols (NE, ; attia2021statistical, ; HUST, ). Data
characteristics under these restricted conditions affect feature extraction
and model design, potentially limiting the success and generalization of their
conclusions. It remains questionable whether these methods perform well under
varied aging conditions. Moreover, focusing on limited aging conditions
restricts the research development of leveraging historical data collected
under different conditions. This limitation separates battery datasets
emphasizing different aging factors as isolated islands, hindering the
development of general modeling approaches.

Here we introduce a deep learning framework, BatLiNet, tailored to predict
battery lifetime across a variety of aging conditions—including electrode
materials, cycling protocols, and temperature fluctuations. At its core, the
framework innovates with ”inter-cell learning,” which contrasts pairs of
batteries to discern lifetime differences, a significant leap from traditional
models that focus solely on individual ”intra-cell” data. This dual approach
not only captures individual degradation patterns but also contextualizes them
within a broader, comparative aging landscape.

Our findings demonstrate that this inter-cell perspective crucially enhances
the model’s predictive precision and robustness, especially under conditions
where data is sparse, such as with novel electrode materials. By integrating
the comparative inter-cell strategy with the conventional intra-cell analysis
into a singular, unified framework, we bridge the gap between isolated and
relative aging scenarios.

The neural networks embedded within BatLiNet adeptly navigate the nonlinear
intricacies inherent to battery aging, ensuring the framework’s adeptness in
learning from limited resources. This adaptability, coupled with our method’s
attention to diverse aging conditions, positions BatLiNet as a comprehensive
solution for accurate, resilient battery lifetime predictions, essential for
advancing reliable energy storage systems.

To validate the effectiveness of BatLiNet, we construct a comprehensive
benchmark by collecting all public datasets (NE, ; CLO, ; SNL, ; HUST, ;
CALCE1, ; CALCE2, ; CALCE3, ) that emphasize different aging conditions and
contain the necessary information to support learning and evaluation. To the
best of our knowledge, this benchmark is the largest and most diverse in terms
of aging conditions for battery lifetime prediction. Our results demonstrate
the remarkable effectiveness of BatLiNet in robustly producing accurate
predictions across diverse aging conditions and in boosting prediction
performance for low-resource conditions. Regarding learning across diverse
aging conditions, our approach exclusively achieves a 10% prediction error
using the first 100 cycles. Moreover, as for learning in low-resource
scenarios, we use $2$ to $8$ cells to achieve $20\%$ prediction error in
predicting the lifetime of new batteries, halving the error of direct learning
on rare data in many cases.

## The BatLiNet Framework

Figure 1. An overview of the BatLiNet framework, where we adopt a lithium-
iron-phosphate battery cell as the target cell and leverage another lithium-
cobalt-oxide battery cell as the reference. a: The feature construction for
intra-cell and inter-cell learning. b: The correlations between constructed
features and prediction labels for both intra-cell (the upper part) and inter-
cell (the lower part) learning. c: The overall pipeline of the BatLiNet
framework.

BatLiNet is a holistic framework that unifies intra-cell and inter-cell
learning. Intra-cell learning involves capturing critical early-cycle patterns
of a single battery cell that highly indicate its lifetime, while inter-cell
learning aims to characterize the differences between a pair of battery cells
in early cycles, implying their lifetime difference. Though serving different
objectives, inter-cell learning can be directly leveraged for battery lifetime
prediction by treating one cell as a target and regarding the other as a
reference, which is assumed to have a known lifetime.

In Figure 1, we provide an overview of the BatLiNet framework. Specifically,
Figure 1a depicts the process of constructing effective features for intra-
cell and inter-cell learning, where we use the voltage-capacity curve during
the discharge stage as a typical example to represent cycle-level features.
Traditional approaches for intra-cell learning operate on a single cell and
obtain effective intra-cell features by calculating the cycle-level feature
differences between an early cycle, such as the 100-th cycle, and an initial
cycle, such as the 10-th cycle (NE, ; attia2021statistical, ). In this work,
we refer to these operations as calculating _intra-cell differences_ , meaning
the feature differences between different cycles within a single cell. To
adapt to diverse aging conditions, we propose to calculate _inter-cell
differences_ by contrasting the cycle-level features between a target cell and
a reference cell in the same cycle. The inter-cell differences help to enhance
the understanding of the relations and differences between different
conditions. Besides, in addition to the discharging voltage-capacity curve,
other cycle-level feature maps, such as the charging voltage-capacity curve,
can also be utilized in both intra-cell and inter-cell learning (detailed in
Methods).

To understand the effectiveness of intra-cell and inter-cell differences
across diverse aging conditions, we visualize the connections between these
difference curves and their corresponding targets for all battery cells in
Figure 1b. In the upper part of Figure 1b, we plot the correlation between the
standard deviation of intra-cell difference curves and the corresponding
lifetime. Similar to the observations in previous studies (NE, ;
attia2021statistical, ), the standard deviation of intra-cell difference
curves for a specific cell roughly presents a near-linear correlation with its
lifetime. However, we additionally observe more scattered points and more
outliers due to the incorporation of much more diverse aging conditions. These
new observations intuitively indicate the inadaptability of past experiences
to different aging conditions. Moreover, we plot the connection between inter-
cell differences and associated lifetime differences in the lower part of
Figure 1b, where all inter-cell differences use the same reference cell.
Similarly, we also observe some near-linear components, non-linear
distributions, and outliers, which call for sophisticated modeling beyond
linear approaches. More importantly, we can find that the relationship between
inter-cell differences and lifetime differences presents a very different
distribution than the corresponding intra-cell relationship, indicating that
learning in intra-cell and inter-cell spaces can be complementary to each
other.

Accordingly, we design a two-branch encoding architecture followed by a shared
linear layer to unify intra-cell and inter-cell learning, as shown in Figure
1c. Given a target cell, we feed its cycle-level features into two separate
branches. The first branch calculates intra-cell difference curves by
comparing features in all early cycles with that of a reference cycle and
employs a specific neural network $f$ for intra-cell learning. In addition,
the second branch includes a reference cell to obtain inter-cell difference
curves and further feeds them into another neural network $g$ for inter-cell
learning. After the encoding procedures of $f$ and $g$, the resulting
representations are mapped to an aligned linear space, allowing us to leverage
a shared linear layer to emit outputs for these two branches (we provide the
derivation for the necessity of this shared linear layer in Methods). During
training, we can randomly assign reference cells to stimulate robust inter-
cell modeling. At inference time, we can randomly select references from
training cells and use the average of outputs from these two branches as the
final prediction. With these customized designs, the BatLiNetframework adapts
to diverse aging conditions and robustly produce accurate battery lifetime
predictions.

## A Comprehensive Benchmark Covering Diverse Aging Conditions

Figure 2. Visualization of datasets. a: The coverage of the dataset employed
in this work significantly surpasses that of previous undertakings, namely the
NE(NE, ), CLO(CLO, ), SNL(SNL, ), HUST(HUST, ), and CALCE(CALCE1, ; CALCE2, ;
CALCE3, ). b: A visualization of the capacity degradation with respect to the
cycles for all the cells. The curves are normalized by their nominal
capacities. The first $1500$ and $100$ cycles are shown. The degradation
patterns are highly non-linear and are difficult to distinguish in the first
$100$ cycles.

To comprehensively evaluate and compare BatLiNetagainst existing modeling
approaches, we curated, to the best of our knowledge, all publicly available
battery data for battery lifetime prediction into five benchmark datasets.

MATR-1, MATR-2, and HUST consist of commercial 18650 lithium-ion
phosphate/graphite (LFP) batteries of the same model. MATR-1 and MATR-2 use
the same training set and evaluate the prediction performance on trained or
unseen charging protocols, respectively. The batteries in HUST employ an
identical charging protocol but different discharge rates to examine the
model’s generalization capability across diverse discharge protocols.

Additionally, we collected battery data used in prior studies including CLO
(CLO, ), CALCE (CALCE1, ; CALCE2, ; CALCE3, ), HNEI (HNEI, ), UL-PUR (ULPUR,
), RWTH (RWTH, ) and SNL (SNL, ). By combining these batteries with MATR-1,
MATR-2 and HUST, we obtained a total collection of 401 batteries and developed
two datasets MIX-100 and MIX-20. MIX-100 examines the typical early prediction
case where models must forecast the $80\%$ end-of-life point using just the
first $100$ cycles of data. MIX-20 poses a more difficult challenge - models
must predict the number of cycles before capacity degrades to $90\%$ of
nominal, using only the first $20$ cycles. Notably, any batteries reaching
end-of-life prematurely during the initial cycles were excluded from the
experiments.

Figure 2a shows that the collected dataset exhibits a more comprehensive
coverage that goes beyond the confines of previous works. Figure 2b further
visualizes the capacity degradation of the collected batteries in the first
$1500$ and $100$ cycles of the batteries. Here the long-term degradation is
highly non-linear and demonstrates considerable variation among batteries. In
contrast, the degradation within the initial $100$ cycles remains
indiscernible for most batteries. Such diverse and intricate characteristics
require the model to capture and distinguish the degradation patterns caused
by various factors for generalization.

## Accurate Battery Lifetime Prediction Empowered by BatLiNet

Method | MATR-1 | MATR-2 | HUST | MIX-100 | MIX-20
---|---|---|---|---|---
RMSE | MAPE(%) | RMSE | MAPE(%) | RMSE | MAPE(%) | RMSE | MAPE(%) | RMSE | MAPE(%)
Training Mean | 399 | 28 | 511 | 36 | 420 | 18 | 573 | 59 | 593 | 102
“Variance” Model(NE, ) | 138 | 15 | 196 | 12 | 398 | 17 | 521 | 39 | 601 | 95
“Discharge” Model(NE, ) | 86 | 8 | 173 | 11 | 322 | 14 | 1743 | 47 | ¿2000 | ¿100
“Full” Model(NE, ) | 100 | 11 | 214 | 12 | 335 | 14 | 331 | 22 | 441 | 53
Ridge Regression(attia2021statistical, ) | 125 | 13 | 188 | 11 | 1047 | 36 | 395 | 30 | 806 | 150
PCR(attia2021statistical, ) | 100 | 11 | 176 | 11 | 435 | 19 | 384 | 28 | 701 | 78
PLSR(attia2021statistical, ) | 97 | 10 | 193 | 11 | 431 | 18 | 371 | 26 | 543 | 77
SVM Regression | 140 | 15 | 300 | 18 | 344 | 16 | 257 | 18 | 438 | 46
Random Forest(attia2021statistical, ) | 140 | 15 | 202 | 11 | 348 | 16 | 211 | 14 | 288 | 31
MLP(attia2021statistical, ) | 162±7 | 12±0 | 207±4 | 11±0 | 444±5 | 18±1 | 455±37 | 27±1 | 532±25 | 61±6
LSTM | 123±11 | 12±2 | 226±36 | 14±2 | 442±32 | 20±1 | 266±11 | 15±1 | 417±62 | 37±7
CNN(attia2021statistical, ) | 115±96 | 9±6 | 237±107 | 17±8 | 445±35 | 21±1 | 261±38 | 15±1 | 785±132 | 41±4
BatLiNet (ours) | 59±2 | 6±0 | 163±12 | 11±1 | 264±9 | 10±1 | 158±7 | 10±0 | 201±18 | 18±1
Table 1. Comparison of prediction errors with baselines. We employ bold font
to emphasize the best-performing method and utilize underlines to denote the
second-best methods. For neural-network-based methods, we report the mean and
standard deviation of eight random seeds. Figure 3. Distribution of prediction
error for deep learning models across five benchmark datasets. Box plots
illustrate the distribution of root mean squared error (RMSE) over eight
random initializations of each deep architecture.

In Table 1, we present the performance comparison of BatLiNet against various
baselines on the five benchmark datasets. The “variance”, “discharge”, and
“full” models (NE, ) employ linear regression on statistical features derived
from discharge capacity-voltage curves. Ridge Regression instead directly fits
a linear model to the raw discharge curves (attia2021statistical, ). Partial
Least-Squares Regression (PLSR) (Geladi1986PartialLR, ) and Principal
Component Regression (PCR) (Shen2009PrincipalCA, ) first group cells based on
their raw discharge curves and then fit linear models on the result groups.
The Support Vector Machine (SVM) (Drucker1996SupportVR, ) and Random Forest
(Breiman2001RandomF, ) are non-linear statistical models with a stronger
fitting ability. For deep models, we compare BatLiNetagainst Multi-layer
Perceptron (MLP)(attia2021statistical, ), the Long-Short Term Memory network
(LSTM)(Hochreiter1997LongSM, ), and Convolutional Networks
(CNN)(Krizhevsky2012ImageNetCW, ).

The ”Discharge” model demonstrates strong performance on MATR-1, MATR-2, and
HUST, yet struggles with diverse chemistries and intricate aging patterns in
MIX-100 and MIX-20. In comparison, Ridge Regression shows lower prediction
error when applied across battery types, indicating that such LFP-focused
features may not effectively generalize to diverse aging conditions. By
explicitly differentiating cells beforehand, PLSR and PCR incorporate
nonlinear complexity into linear frameworks and further improve over Ridge
Regression. Both SVM and RF excel on MIX-100 and MIX-20 compared to other
linear baselines, confirming that the complex degradation patterns call for
models with stronger fitting capability.

The deep learning models exhibit high variability in performance depending on
random initialization. As shown in Figure 3, the error distribution of the
deep models fluctuates widely across random seeds. The MLP architecture
demonstrates relatively modest prediction errors with the lowest variance on
all datasets. In contrast, CNN displays the largest variance in performance,
yet achieves the lowest errors on three datasets among baselines given optimal
seeds (attia2021statistical, ). LSTM strikes a favorable balance between
accuracy and variance. However, all deep models are susceptible to overfitting
and exhibit minimal predictive advantage compared to conventional statistical
approaches regarding both regression accuracy and error variance.

While no single technique consistently optimizes performance across datasets,
BatLiNetachieves $\leq 11\%$ mean absolute percentage error (MAPE) given the
first $100$ cycles and $\leq 18\%$ for the first $20$ cycles. This reduces the
root mean square error (RMSE) versus the top baseline by $31.4\%$, $5.8\%$,
$18.0\%$, $30.2\%$, and $25.11\%$ on the five datasets, respectively. By
jointly training the intra-cell and inter-cell branches, BatLiNetcombines
strong fitting capabilities beyond conventional statistical models with
robustness to random initialization lacking in existing deep models. Through
this blend of strengths, BatLiNetdelivers accurate and reliable lifetime
predictions across diverse battery aging conditions.

## Facilitating Effective Learning in Low-resource Scenarios

Figure 4. Predictive performance of three learning paradigms that employ
LFP/graphite cells to benefit learning in low-resource scenarios. We report
the mean and variance of the percentage error under eight random seeds.

In practice, battery development is often constrained by limited resources,
resulting in a restricted number of available test batteries. To develop an
accurate life prediction model, cycling a considerable number of batteries to
end-of-life is required for label collection. However, this process is both
time-consuming and unsustainable. An alternative methodology is to leverage
the abundant historical data of batteries with varying aging states to augment
model training under such low-resource scenarios.

To simulate resource-constrained applications, the cells in MIX-100 were
sorted by cathode material, and the 275 LFP/graphite cells were leveraged to
augment the prediction performance on the $37$ LCO cells, $22$ NCA cells, and
$69$ NMC cells. Among the test batteries, $21$ LCO cells, $14$ NCA cells, and
$53$ NMC cells were randomly sampled for model evaluation, and we employ $1$,
$2$, $4$, $8$, and $16$ cells from the remainder to simulate varying cycling
test budgets. Three learning paradigms were investigated to comprehensively
analyze the influence of historical data: 1) direct training of a CNN on the
target cells, 2) transfer learning where a CNN was pre-trained on LFP cells
then fine-tuned on the target cells, and 3) training BatLiNeton the combined
LFP and target cells.

Figure 4 presents the performance of the three learning paradigms under
resource-constrained conditions. Pre-training the model on historical
LFP/graphite cell data substantially reduced prediction error by $10\%$ for
NCA cells. However, it resulted in inferior predictions for LCO and NMC cells,
suggesting that degradation patterns vary between battery chemistries. In
contrast, BatLiNetleveraged the LFP cells to improve over direct learning for
all three cathode materials, indicating the inter-cell modeling between LFP-
target pairs generalizes across cathode materials. Notably, BatLiNetfurther
reduced prediction error by $10\%$ over transfer learning, achieving $20.26\%$
MAPE using only two to eight target cells and historical LFP data. This high
data efficiency greatly mitigates the data hunger in battery development and
can substantially reduce downstream application modeling costs such as
developing novel electrode materials or electrolytes. In contrast, Attia et
al. (CLO, ) developed a linear model trained on a dataset of $41$ LFP cells
when optimizing the rapid charging protocol specifically for batteries
possessing the same lithium iron phosphate cathode chemistry.

## Working Mechanisms of BatLiNet

Figure 5. Ablation studies to illustrate the mechanism of BatLiNet. a:
Leveraging the joint training of intra-cell and inter-cell branches,
BatLiNetachieves superior performance than the ensemble of separately trained
inter-cell and intra-cell branches. b: Enhanced predictive performance
necessiates the inclusion of the complete voltage and current signals, as well
as their interactions, from both charging and discharging stages.

Here we conduct further ablation experiments to study the mechanism of
BatLiNet.

Figure 5a compares the predictive performance of different modeling branches
in BatLiNet. Across all datasets, intra-cell models exhibit the highest error
and variance. The inter-cell model shows good robustness with lower variance
across all datasets, yet its prediction accuracy is slightly lower than that
of the intra-cell modeling on some datasets, as its predictions are implicitly
derived through the reference cells. The ensemble of the two separately
trained branches obtains better performance on HUST and MIX-100, yet on other
datasets it inherits the flaws of either branch, resulting in higher variance
or suboptimal accuracy. BatLiNetachieved the most robust and precise lifetime
estimation across all datasets, demonstrating that through the co-training
with shared prediction layer, the model better integrates the strength of both
branches and yields the strongest predictive performance.

Figure 5b displays the performance of BatLiNetwhen using a subset of the six
capacity-indexed features. It’s evident that all six features play a crucial
role in achieving improved predictive performance. The current-based features
can lead to high variance and large errors, yet they are indispensable for the
extreme early prediction scenarios in MIX-20. The voltage- and discharge-based
features demonstrate strong performance across all datasets. Merging voltage
and current features from both charge and discharge stages results in enhanced
performance, with the best outcomes achieved by employing all six features.

## Conclusion

By introducing inter-cell learning and unifying it with intra-cell learning,
our framework BatLiNet has significantly boosted the performance of battery
lifetime prediction across diverse aging conditions. It is noteworthy that the
proposed inter-cell learning, enabling the modeling of the relations and
differences among diverse aging conditions, not only provides exclusive yet
complementary value for traditional intra-cell learning but also facilitates
the effective knowledge transfer from rich conditions with abundant historical
data to low-resource scenarios of emerging needs. Looking into the future, our
method can be leveraged in various developing aspects that lead to varied
aging conditions, such as different fast charging protocols
(zeng2023extreme_fast_charging, ) and new electrode materials
(shraer2022dev_pos_material, ). Specifically, as an accurate and robust
lifetime predictor across different aging conditions, our method holds the
largely improved capability in accelerating the development and optimization
of lithium-ion batteries (CLO, ). Moreover, the idea of modeling inter-cell
differences can be extended to other crucial prediction tasks, such as
predicting the state of charge and health (ng2020soc_soh_nmi, ;
Roman2021NMI_soh_pipeline, ), broadly benefiting battery management and
applications.

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## Methods

### Building Cycle-level Feature Maps

We have shown how to build features for intra-cell and inter-cell learning in
Figure 1a, by leveraging the discharging voltage-capacity curve as an example
for cycle-level features. Here we include a comprehensive discussion about the
procedure of building cycle-level feature maps.

The raw data available for each cycle includes various electrical signals
collected during the charging and discharging process. However, there are
several obstacles preventing us from directly utilizing these raw signals.
First, the electrical signals in different cycles usually have variable
lengths. Hence, it is necessary to align them properly into a regular format
to fit for neural networks. Meanwhile, due to the fine-grained recording of
electrical signals, the raw high-dimensional features are highly redundant and
require effective noise reduction. Besides, each cycle contains various
electrical signals (voltage, current, states of charge/discharge) on two
different stages (charge, discharge), which call for effective representations
that cannot only contain valuable electrical signals but also yield the
ability to differentiate respective values of various signal types within the
same or between different stages.

With full consideration of these obstacles, we develop specific processing
steps to obtain cycle-level feature maps from the raw electrical signals. The
core idea is to use the normalized state of charge ($Q$) as a new index to
align different electrical signals in different cycles and compute their
differentiation within the same stage or between charge and discharge stages.
In this way, we do not need to worry about the variable lengths across cycles.
Besides, due to the redundant recording of electrical signals, we can obtain
these $Q$-indexed series by performing interpolations on raw time-indexed
signals and easily control the feature dimensions by adjusting interpolation
granularity. After this step, we can obtain four types of processed series
indexed by $Q$, $V_{c}(Q)$, $V_{d}(Q)$, $I_{c}(Q)$, and $I_{d}(Q)$, where $Q$
varies from 0 to 1 with a pre-specified step. We further derive two additional
signals to characterize the connections between the charge and discharge
stages: 1) $\Delta V(Q)=V_{c}(Q)-V_{d}(Q)$, meaning the gap between charge
voltages and discharge voltages, and 2)
$R(Q)=(V_{c}(Q)-V_{d}(Q))/(I_{c}(Q)-I_{d}(Q))$, corresponding to the status of
internal resistance as $Q$ varies. These cycle-level features serve as the
basic elements for calculating intra-cell and inter-cell differences.

There are two critical processing steps to ensure the effectiveness of
calculating these differences from cycle-level feature maps. This first is to
normalize the capacity of all batteries by their respective norminal
capacities, ensuring the range of $Q$ falls within $[0,1]$, to adapt to
various battery types with very different nominal capacities. The second is to
eliminate the data noise introduced by abrupt changes in current and voltage
signals due to the variations in cycling protocols. We employ a rolling-
median-based filter to alleviate this issue:

(1) $\displaystyle\mathbf{M}$
$\displaystyle=\text{rolling\\_median}(\mathbf{r},w),$ (2)
$\displaystyle\Delta\mathbf{M}$
$\displaystyle=\text{abs}(\text{rolling\\_median}(\mathbf{M},w)),$ (3)
$\displaystyle r_{t}^{new}$ $\displaystyle=\begin{cases}M_{t}&\text{if }\Delta
M_{t}>3\times\text{median}(\Delta\mathbf{M}),\\\
r_{t}&\text{otherwise.}\end{cases}$

Here $\mathbf{r}$ is the raw signal and $w$ represents the window size of the
rolling operations.

### Unifying Intra-cell and Inter-cell Learning

Given the intra-cell feature of a battery cell, denoted as $\bm{x}$, and its
cycle life $y$, we want to optimize the following objective to obtain a
perfect cycle life predictor.

(4) $\min_{\bm{\theta}}\mathbb{E}_{(\bm{x},y)\in D}\left\lVert
f_{\bm{\theta}}(\bm{x})-y\right\rVert_{2}^{2},$

where $D$ denotes the data distribution, $f_{\bm{\theta}}$ is an encoding
function parameterized by ${\bm{\theta}}$, and $\mathbf{w}_{f}$ is the weight
of the last prediction layer. In practice, we need to perform the empirical
risk minimization over limited training instances, that is

(5) $\min_{\bm{\theta}}\sum_{i=1}^{N}\left\lVert
f_{\bm{\theta}}(\bm{x}_{i})-y_{i}\right\rVert_{2}^{2}.$

However, battery lifetime prediction is a special task that suffers from the
data-scarce challenge due to the necessity of non-linear modeling and the huge
cost of obtaining data labels. In this scenario, instantiating
$f_{\bm{\theta}}$ as a neural network has a high risk of overfitting. To
alleviate the data-scarce issue, we consider modeling the differences between
two distinct cells:

(6) $\min_{\bm{\theta}}\mathbb{E}_{(\Delta\bm{x},\Delta y)}\left\lVert
g_{\bm{\phi}}(\Delta\bm{x})-\Delta y\right\rVert_{2}^{2},$

where $\Delta\bm{x}=\bm{x}-\bm{x^{\prime}}$, $\Delta y=y-y^{\prime}$,
$(\bm{x},y)$ and $(\bm{x^{\prime}},y^{\prime})$ are independently sampled from
$D$, and $g_{\bm{\phi}}$ is a function parameterized by ${\bm{\phi}}$ that
operates in the space of $\Delta\bm{x}$.

The inter-cell learning formulation (6) assumes that the differences in
feature representations for any pair of battery cells hold a unified
relationship with the differences in their lifetimes. Specifically, we can
establish a clear connection between $f_{\bm{\theta}}$ and $g_{\bm{\phi}}$ in
the linear setting. For example, if $y$ is zero-centered (can be done via pre-
processing), and the optimal solution for the original objective (4) is
$f_{{\bm{\theta}}^{*}}(\bm{x})={\bm{w}^{*}}^{T}\bm{x}$, then it is easy to
verify that $g_{{\bm{\phi}}^{*}}=f_{{\bm{\theta}}^{*}}$ is also the optimal
solution for the contrastive objective (6) because
$\mathbb{E}_{(\Delta\bm{x},\Delta
y)}\left\lVert{\bm{w}^{*}}^{T}\Delta\bm{x}-\Delta y\right\rVert_{2}^{2}$ can
be decomposed into

(7) $\mathbb{E}_{(\bm{x},y)\in
D}\left\lVert{\bm{w}^{*}}^{T}\bm{x}-y\right\rVert_{2}^{2}+\mathbb{E}_{(\bm{x}^{\prime},y^{\prime})\in
D}\left\lVert{\bm{w}^{*}}^{T}\bm{x}^{\prime}-y^{\prime}\right\rVert_{2}^{2}.$

In the non-linear setting, such as using neural networks as function
approximators, it is intractable to establish the exact connection between
$f_{\bm{\theta}}$ and $g_{\bm{\phi}}$, which leads to two separate
optimization procedures. Inspired by the same optimality of the objectives (4)
and (6) under the linear setting, we propose to share the last linear layer of
$f_{\bm{\theta}}$ and $g_{\bm{\theta}}$ when using neural networks, that is

(8) $f_{\bm{\theta}}(\bm{x})=\bm{w}^{T}h_{\bm{\theta}}(\bm{x}),\quad
g_{\bm{\phi}}(\Delta\bm{x})=\bm{w}^{T}h_{\bm{\phi}}(\Delta\bm{x}),$

where $\bm{w}$ is the shared parameter, $h_{{\bm{\theta}}}(\cdot)$ and
$h_{{\bm{\phi}}}(\cdot)$ are two separate neural networks parameterized by
$\bm{\theta}$ and $\bm{\phi}$, respectively. In this way, we can connect the
optimization of the objectives (4) and (6) via the shared parameter $\bm{w}$
and enjoy the complementarity between the initial intra-cell learning and the
new inter-cell modeling.

Moreover, we also need to perform empirical risk minimization to mimic the
desired objective (6). Specifically, given that we only have $N$ training
instances, we use $N(N-1)$ pairs of instances to substitute the expectation
over independently sampled instance pairs. Together with the original
regression, we have the following joint empirical risk minimization problem:

(9)
$\min_{\bm{w},{\bm{\theta}},{\bm{\phi}}}\sum_{i=1}^{N}\left\lVert\bm{w}^{T}h_{\bm{\theta}}(\bm{x}_{i})-y_{i}\right\rVert_{2}^{2}+\lambda\sum_{i=1}^{N}\sum_{\begin{subarray}{c}j=1\\\
{j\neq
i}\end{subarray}}^{N}\left\lVert\bm{w}^{T}h_{\bm{\phi}}(\Delta\bm{x}_{i,j})-\Delta
y_{i,j}\right\rVert_{2}^{2},$

where $\Delta\bm{x}_{i,j}=\bm{x}_{i}-\bm{x}_{j}$, $\Delta
y_{i,j}=y_{i}-y_{j}$, and $\lambda$ is a hyper-parameter to balance two
regression objectives.

After the optimization stage, we can leverage the neural networks for encoding
intra-cell and inter-cell differences to make predictions. Given an unseen
instance $\bm{x}$, we have

(10)
$\hat{y}^{o}=\bm{w}^{T}h_{\bm{\theta}}(\bm{x}),\quad\hat{y}^{c}=\bm{w}^{T}h_{\bm{\phi}}(\bm{x}-\bm{x}^{\prime})+y^{\prime},$

where $\hat{y}^{o}$ and $\hat{y}^{c}$ are the predictions in the original and
contrastive space, respectively, and $(\bm{x}^{\prime},y^{\prime})$ can be
sampled from the training instances. Last, we predict the lifetime as
$\hat{y}=\alpha\hat{y}^{o}+(1-\alpha)\hat{y}^{c}$, where $\alpha$ is a hyper-
parameter to balance the two types of predictions.

### Selection strategy for the reference cells

Figure 6. Selection strategy for reference cells and the impact of the number
of reference cells for each battery. a: absolute prediction error for each
battery using the optimal, worst, and median of the reference cells. The
shades represents the variance among eight random seeds. b: a plot of
prediction error with respect to different number of the reference cells. c:
visualization of the inference speed and memory consumption with different
number of reference cells.

While the reference signals of intra-cell models are restricted to their own
cycles, inter-cell modeling allows flexible selection of reference batteries.
This additional freedom enables us to enhance the predictive performance
through the careful choice of reference cells.

Figure 6a demonstrates the substantial impact of different reference cell
selection strategy on the prediction error for each battery. The gap between
the least favorable and optimal reference choices can lead to an absolute
error exceeding $1500$ cycles. The rules for the optimal reference choice of
reference varies among the target batteries, as shown in Figure S6. In
BatLiNet, we employ the median of the predictions of a reference battery group
of size 32 as the final prediction, which runs at 353 target batteries per
second during inference. Using this reference ensemble strategy, both the mean
and variance of the inference error decrease as the number of reference
batteries increases. Impressively, this choice leads to a significant
improvement compared to the worst-case scenario, resulting in an absolute
error reduction of more than $1000$ cycles for specific cells. However, a
substantial gap still exists between the median choice and the optimal one.
Closing this gap requires the model to intelligently select the most suitable
reference cell for each battery.

Figure 6b illustrates the relationship between the number of reference cells
and the prediction error of InterCD. As the number of reference cells
increases, both the average prediction error and the variance between
different random seeds decrease. This indicates that a larger number of
reference cells simultaneously enhances the model’s accuracy and robustness.

Figure 6c depicts the impact of the reference cell count on the inference
speed and batch size for BatLiNet. With an increase in reference size, there
is a corresponding rise in inference time and a reduction in batch size. A
large number of reference cells leads to higher computational cost, where the
number of batteries processed in parallel decreases from $512$ to $8$.
However, even with a very large reference size of $64$ for each cell, the
inference speed of the model remains notably fast, operating at $178.8$ cells
per second. This underscores the capability of BatLiNetas a real-time battery
lifetime predictor.

### Network architecture design and experimental setups

The input features $\mathbf{x}$ have dimensions of $(B,6,H,W)$, where $B$
represents batch size, $H$ stands for cycle count, and $W$ denotes the
dimension of interpolated normalized capacity. The encoder contains two 2D
convolutional layers, each followed by an average pooling layer and a ReLU
activation layer. Different from computer vision applications, our use of
average pooling allows the model to focus on the overall characteristics of
the difference curves, rather than abrupt changes in specific locations such
as spikes caused by data misalignment. The end of the encoder is a fully
connected layer that compresses the spatial dimensions $(H,W)$ to $1$,
facilitating subsequent predictions through the shared linear regression
layer.

We employ a hidden layer dimension of $32$ across all models. For all neural
network models, we conduct eight runs using random seeds ranging from $0$ to
$7$ to evaluate their performance under different initialization conditions.
In each experiment, $32$ reference cells are employed for inference and we
take the median of these reference cells as the final prediction for the
InterCD branch. All experiments are executed on a server with 8 GTX 3090 GPUs,
and we leverage PyTorch’s timer to conduct repeated runtime assessment.

### Evaluation metrics

To comprehensively evaluate model performance, this work utilizes three
standard error metrics: root mean square error (RMSE), absolute percentage
error (APE), and mean absolute percentage error (MAPE). The RMSE provides an
absolute measure of the difference between predicted values ($\mathbf{y}$) and
ground truth labels ($\hat{\mathbf{y}}$). It is calculated by first finding
the squared error between each paired prediction and label, summing these,
then taking the square root of the average squared error across all samples
$N$:

(11)
$\mathrm{RMSE}(\mathbf{y},\hat{\mathbf{y}})=\sqrt{\frac{1}{N}\sum_{n=1}^{N}(y_{n}-\hat{y}_{n})^{2}}$

Lower RMSE values indicate better model performance, with a value of 0
corresponding to perfect predictions. However, RMSE lacks normalization and
thus can be difficult to interpret across datasets. APE provides a relative
percentage measure of the deviation between each prediction-label pair. It is
defined as the absolute ratio between the error and the ground truth label:

(12)
$\mathrm{APE}(y_{n},\hat{y}_{n})=\left|\frac{y_{n}-\hat{y}_{n}}{\hat{y}_{n}}\right|$

Unlike RMSE, APE is unitless and normalized, making it more interpretable
across tasks. However, it lacks a global view of performance across samples.
Finally, MAPE reports the mean of APE across all predictions to quantify
overall performance. It is calculated by taking the average absolute
percentage error:

(13)
$\mathrm{MAPE}(\mathbf{y},\hat{\mathbf{y}})=\frac{1}{N}\sum_{n=1}^{N}\mathrm{APE}(y_{n},\hat{y}_{n})$

Lower MAPE indicates better generalization. Together, these metrics provide
comprehensive and complementary insights into model performance.

### Cell-level error analysis

Figure 7. Cell-level error analysis. a: Cell-level error comparison between
BatLiNetand baselines on MIX-100. b-f: plots of predicted lifetime compared
with true lifetime on the five benchmark datasets.

Both RMSE and MAPE reflects the average error on the dataset scale, which can
conceal anomalously large errors in individual samples. Here we study the
error distribution on the cell-level to provide a more granular view of the
predictive performance of BatLiNet.

Figure 7a plots the cumulative absolute error with respect to the fraction of
test samples in MIX-100 dataset. For each method, the test cells are added in
the same order. The error curves of the random forest (RF) and
BatLiNetinitially coincide for the first approximately $50\%$ of test samples,
after which the error of RF begins to escalate at a faster rate than that of
BatLiNet. This suggests that RF accurately captures the patterns within a
subset of the samples, whereas BatLiNetexcels at capturing the patterns in the
remaining portion. The error curve slopes of other baselines are all smaller
than BatLiNet, indicating that BatLiNetconsistently outperforms these methods
for each data point.

Figure 7b-f shows the predicted lifetimes by BatLiNetversus the ground truth
on the five benchmarks. The errors on MATR-1 are well controlled across all
batteries, especially for low lifetime batteries where the errors are far
below $10\%$. On the MATR-2 dataset, the majority of batteries have errors
within $20\%$, but the model tends to underestimate high lifetime batteries
due to distribution shift between training and testing. The errors on the HUST
dataset are mainly from severely underestimating an outlier, which can greatly
inflate the average error as the number of test samples is small. On the
MIX-100 dataset, $84.7\%$ of batteries have prediction errors within $20\%$,
with only a few outliers among the 137 test samples. This shows that
BatLiNetcan accurately capture complex battery degradation patterns across
varying battery types and cycling conditions, leading to highly accurate and
robust lifetime predictions in most cases. The model still struggles on
MIX-20, where many samples have large prediction biases, including both over
and under-estimation. Accurate lifetime prediction remains a challenging
problem in such information-scarce scenarios.

### Data collection and benchmark construction

We collect the batteries that are applicable to lifetime prediction from
existing data sources. The data batteries contain the voltage and current
signals during the entire cycling test. Table S1 shows the statistics of the
batteries collected from different data sources. We collected the CS2 and CX2
commercial prismatic cells from the Center for Advanced Life Cycle Engineering
(CALCE). For all cells, an identical charging profile was employed, utilizing
a standard constant current/constant voltage procedure with a consistent
current rate of 0.5C until the voltage reached 4.2V. Subsequently, the voltage
was maintained at 4.2V until the charging current fell below 0.05A. The
discharge cut-off voltage for all batteries was set at 2.7V. The data for all
cells are publicly available.

The 124 commercial lithium-ion batteries of MATR-1 and MATR-2 were cycled to
failure under fast-charging conditions. Specifically, these lithium-ion
phosphate (LFP)/graphite cells, developed by A123 Systems (APR18650M1A), were
cycled in horizontal cylindrical fixtures using a 48-channel Arbin LBT
potentiostat. The experiments occurred within a temperature-controlled chamber
with forced convection, set at 30°C. The cells have a nominal capacity of 1.1
Ah and a nominal voltage of 3.3 V. The cells were subjected to a two-stage
charging protocol, involving a rapid charge rate until reaching 80% capacity,
followed by a charge at 1C. The upper and lower cutoff potentials were set at
3.6 V and 2.0 V, respectively, and all cells underwent discharge at 4C. The
data for all cells are publicly available.

We collected additional 55 LFP/graphite cells that are used in Ref. CLO. All
cells underwent an optimized six-step, 10-minute fast-charging protocol. Other
specifications, including the battery model, environment temperature, and
discharge protocols are the same as MATR-1 and MATR-2. The data for all cells
are publicly available.

RWTH dataset comprises time-series electrical and temperature signals from a
cyclic aging test involving 48 lithium-ion battery cells. During the
experiment, 48 cells of identical configuration underwent aging using the same
profile under uniform conditions. The cells utilized are Sanyo/Panasonic
UR18650E cylindrical cells, commercially available and mass-produced through a
well-established fabrication process, which is designed with a carbon anode
and NMC as the cathode material. The data for all cells are publicly
available.

UL-PUR dataset consists of 10 commercial 18650 cells with a graphite negative
electrode and an NCA positive electrode. The cells were cycled at 0.5C at
2.7-4.2V (0-100% SOC) or 2.5-96.5% SOC at room temperature, until experiencing
a 20% capacity fade. The HNEI dataset comprises 14 commercially available
18650 cells featuring a graphite negative electrode and a composite positive
electrode made up of NMC and LCO. These cells were cycled at 1.5C to achieve a
depth of discharge of 100% for over 1000 cycles at room temperature. The SNL
dataset consists of 61 commercial 18650 NCA, NMC, and LFP cells cycled to 80%
capacity. These cells experience different temperature, depth of discharge,
and discharge current on the long-term degradation process. These three
datasets currently are available at Battery Archive upon request.

Table 2 and 3 shows the statistics of the data sources and our organized
datasets. The code for organizing and preprocessing the datasets are publicly
available.

Table 2. Statistics of data sources used in this study. |  | Cell
---
count
| Electrode
---
materials
| Charge
---
protocols
| Discharge
---
protocols
| Environment
---
temperatures
| Package
---
structures
CALCE | 13 | 1 | 1 | 2 | 1 | 1
HNEI | 14 | 1 | 1 | 1 | 1 | 1
HUST | 77 | 1 | 1 | 77 | 1 | 1
MATR-1 | 84 | 1 | 61 | 1 | 1 | 1
MATR-2 | 81 | 1 | 47 | 1 | 1 | 1
CLO | 45 | 1 | 9 | 1 | 1 | 1
RWTH | 48 | 1 | 1 | 1 | 1 | 1
SNL | 61 | 3 | 1 | 4 | 3 | 1
UL-PUR | 10 | 1 | 1 | 1 | 1 | 1
Total | 401 | 5 | 83 | 85 | 5 | 2
Table 3. Statistics of the organized datasets used in this study. Note that we filtered out batteries that is not reached their end of life during the early cycles. |  | Cell
---
count
| Train cell
---
count
| Test cell
---
count
| End of life
---
percentage
| Early cycle
---
count
| Max
---
lifetime
| Min
---
lifetime
MATR-1 | 83 | 41 | 42 | 80% | 100 | 2237 | 300
MATR-2 | 81 | 41 | 40 | 80% | 100 | 2160 | 300
HUST | 77 | 55 | 22 | 80% | 100 | 2691 | 1140
MIX-100 | 342 | 205 | 137 | 80% | 100 | 2691 | 148
MIX-20 | 354 | 207 | 147 | 90% | 20 | 2323 | 104


# Monolithic atom interferometry

Johannes Fiedler
Department of Physics and Technology
University of Bergen,
Bergen, Norway
<EMAIL_ADDRESS>
&Kim Lefmann
Niels Bohr Institute
University of Copenhagen
København, Denmark
&Wolf von Klitzing
Institute of Electronic Structure and Laser
Foundation for Research and Technology-Hellas
Heraklion, Greece
&Bodil Holst
Department of Physics and Technology
University of Bergen,
Bergen, Norway
<EMAIL_ADDRESS>
###### Abstract

Atom and, more recently, molecule interferometers are used in fundamental
research and industrial applications. Most atom interferometers rely on
gratings made from laser beams, which can provide high precision but cannot
reach very short wavelengths and require complex laser systems to function.
Contrary to this, simple monolithic interferometers cut from single crystals
offer (sub) nano-meter wavelengths with an extreme level of stability and
robustness. Such devices have been conceived and demonstrated several decades
ago for neutrons and electrons. Here, we propose a monolithic design for a
thermal-beam molecule interferometer based on (quantum) reflection. We show,
as an example, how a reflective, monolithic interferometer (Mach-Zehnder type)
can be realised for a helium beam using Si(111)-H(1$\times$1) surfaces, which
have previously been demonstrated to act as very robust and stable diffractive
mirrors for neutral helium atoms.

##  1 Introduction

The field of atom interferometry has expanded enormously over the last few
decades. Atom interferometers are used in various applications, from magnetic
and gravity sensing [1, 2], quantum metrology [3] to atomic clocks [4]. They
may even be used as dark matter and gravitational wave detectors [5] also in
space [6, 7]. Compact, portable atom gravimeters for prospecting, oil survey
and geophysical investigations have recently become commercially available
[8]. Atom interferometers will also be useful as accelerometers for sub-sea
navigation in submarines and, more recently, underwater drones [9, 2]. This,
however, will require very compact solutions, which are not presently
available.

Atom interferometers use either cold atoms (including Bose-Einstein
Condensates) [10] or thermal atoms beams [11], and more recently hot thermal
vapours [12]. Most optical interferometers have, by now, been realised as atom
interferometers, including Young’s double slit, Mach-Zehnder, Talbot-Lau,
Ramsey-Bordé and Sagnac interferometers.

Historically, Young’s double slit makes the simplest atom interferometer. The
beam is split into two paths by passing through a double slit, and the
interference pattern is observed on a screen further down the beam path. It
was realised for atoms for the first time in 1991 using metastable helium
atoms passing through a thin gold foil [13].

The simplest split-path atom interferometer is arguably the Mach-Zehnder
interferometer. It exploits the de Broglie wavelength of the atoms in a
diffraction grating configuration with split beam paths. The first Mach-
Zehnder atom interferometer was realised in 1991 [11] using a sodium beam and
solid transmission diffraction gratings. Later in 1995, it was developed
further by using metastable neon and argon and transmission diffraction
gratings made of standing light waves [14, 15], in 2002 using ground-state
lithium also with light-wave gratings [16] and later again using neutral
helium with solid gratings. Results from the last mentioned instrument were
never published, but it is mentioned in a review paper from 2009 [17].

In the Talbot-Lau interferometer, the self-imaging property of a grating is
exploited in near-field diffraction. The atom paths are not truly separated;
therefore, this type of interferometer has been used extensively for
experiments with heavy molecules where the de-Broglie wavelength is very
small. The first Talbot-Lau atom interferometer was realised in 1994 [18].

Where the Mach-Zehnder interferometer and the Talbot-Lau interferometers are
adapted from light optics, the Ramsey-Bordé interferometer, first realised in
1949 by Norman Ramsey [19], can only be used for atoms: the principle is
diffraction by absorption of a single photon on a weakly allowed transition to
split the wave package. In the 1980th, this interferometer has been further
developed by Christian Bordé by using atomic recoil to create a beam splitter
[20]. This interferometer type is currently the standard for high-precision
measurements, such as atomic clocks.

In light optics, the Sagnac interferometer, also called ring interferometer,
relies on a beamsplitter mirror to create two beams that travel equidistant
paths in opposite directions through a ring structure guided by reflective
mirrors. The two beams meet at the starting point, where they interfere and
are made to exit the ring. The first atom interferometer using the Sagnac
effect was realised in 1991 using a Ramsey–Bordé configuration of a state-
labelled atom interferometer based on single-photon transitions, with a beam
of atoms traversing two pairs of travelling wave fields. The laser fields
within each pair are separated by a distance $D$, while the two pairs are
separated by $d$ and are counter-propagating with respect to each other [21].
By rotating the interferometer, the counter-propagating beams collect
different phases along their optical paths leading to an interference pattern
on the screen. Such a configuration provides an absolute measurement of the
rotational speed.

The atomic structure of a single crystal offers a simple periodic diffractive
grating. Thus, it could produce many different types of interferometers, where
the monolithic construction guarantees extreme stability. Interferometers
based on transmission through solid slabs of material have been demonstrated,
e.g. X-rays [22, 23], neutrons [24] and electrons [25]. Unfortunately, these
techniques are inapplicable to atoms, which interact too strongly with any
solid material they travel through. Monolithic atom interferometers have been
used widely in neutron scattering experiments observing gravitationally
induced interference (in transmission) [26] and the quantised states of
neutrons in the presence of gravitational fields with perfectly reflecting
mirrors [27]. Neutrons are sensitive to external forces and, thus, suitable
candidates for quantum sensing. However, such experiments require an
extensive, costly infrastructure to create, control and detect the neutron
beam. This also applies to cold atom interferometers. Thermal atom beams are
easier to create and couple more robust to external fields due to the higher
mass of the atoms. A further advantage of thermal atom interferometers is that
they can operate continuously, dramatically decreasing the temporal
resolutions.

Here, we propose a novel interferometer based on the reflection of atoms on
monolithic single-crystal structures. The basic operation principle is
depicted in Fig. 1: an incident beam of atoms is reflected by the crystal
lattice (A) into two components, which impinge onto a second mirror and
recombine on the third reflection.

In the past, atoms had been neglected, largely because the atoms most commonly
used in interferometry (Rubidium, Rb [28]; Caesium, Cs [29]; Argon, Ar [30];
Sodium, Na [31]; Potassium, K [32]) will stick to surfaces under most
conditions. Similarly, metastable atoms, which have also been used for
interferometry (Argon, Ar [33]; Helium, He [13]), will decay upon impingement.
A further practical challenge for a reflection-based interferometer is the
contamination of the reflecting surface, which distorts the diffraction. For
example, all metal surfaces will be covered in physisorbed molecules within
hours, even in ultra-high vacuum [34, 35, 36].

Noble gasses, including ground-state helium, H2, HCl and other molecules, are
known to scatter from various surfaces over a broad temperature range without
sticking to them [37, 38, 39]. Over the last years, focusing mirrors for
neutral, ground-state helium have been developed for neutral helium
microscopes [40]. An important requirement for these mirrors is that they must
remain stable in a vacuum for months. One of the solutions implemented was
Si(111)-H$(1\times 1)$ [41]. Detailed experiments on He and H2 scattering were
performed [42, 43] and the interaction potential between Helium and
Si(111)-H$(1\times 1)$ calculated [43]. This interaction potential was then
used to obtain the intensity of the different diffraction peaks for a range of
conditions [44].

The advantage of the Si(111)-H$(1\times 1)$ surface from an experimental point
of view is that it can be prepared chemically by dipping the Si(111) crystal
in an HF solution [45]. This means a monolithic configuration with two
reflecting surfaces facing each other can be fabricated at any spacing. The
additional advantage of using the Si(111)-H$(1\times 1)$ surface is the small
lattice constant of $a_{\rm S}=3.383$ Å [42] which, together with the
wavelength of, as an example, helium atoms in a room temperature beam:
$\lambda_{\rm dB}=0.55\,\rm{\AA}$, ensures a very big wave-package separation.
Recent matter-wave interferometers typically split the wave package over a few
milliradian [46, 47, 33, 48]. In contrast, using the room-temperature helium
beam described above, the proposed new interferometer splits the matter wave
over 0.5 radians.

The atom interferometer we introduce here uses reflective atom-surface
diffraction as a beam splitter. Further reflections from a parallel surface
yield the recombination of the wave and thus the interference; see Fig. 1. We
present a theoretical model determining the expected interference patterns and
apply the model to the interference of helium atoms using Si(111)-H$(1\times
1)$ surfaces, where we concentrate on describing the general principles by
describing an ideal system with a perfectly coherent and monochromatic beam
and an experimentally based model for the diffraction probabilities. We have
chosen an experimentally realisable parameter set providing all possible
superpositions occurring in such interferometer: single-path transmission,
double-path superposition with vanishing phases and multipath interference.
Finally, we discuss how a reflective interferometer based on quantum
reflection can be addressed. The paper finishes with a conclusion and outlook
on future work.

Figure 1: Sketch of the optical paths within a monolithic, reflection
interferometer: the beam is reflected three times between the surfaces of two
parallel slabs (grey area) separated by the distance $s$. The incoming beam is
reflected at point A with an incidence angle $\alpha$. Two different
diffraction orders are selected: Reflection towards point B with diffraction
angle $\beta$ and reflection towards B′ with diffraction angle $\gamma$. At
point B, the incidence angle is the same as the outgoing angle in point A:
$\beta$. Part of this beam is reflected towards point C with diffraction angle
$\delta$. At point B′, the incidence angle is given by $\gamma$ due to the
reflection at point A and reflected at the diffraction angle $\varepsilon$. In
point C, the incoming waves with incidence angles $\delta$ and $\varepsilon$
are recombined, leaving the slab with a reflection angle $\zeta$.

##  2 The reflective interferometer

###  2.1 Geometric arrangement

The general arrangement of a monolithic reflection interferometer is depicted
in Fig. 1. A slab is cut into a U-shaped monolith to form two parallel planar
surfaces with a distance $s$ being sufficiently large to achieve propagating
waves inside the interferometer. A particle beam will be diffracted minimally
three times at points A, B and C. The beam will be split at point A, and each
part will be reflected at point B and recombined in point C, where they
interfere.

In detail: a particle beam is sent via an incidence angle $\alpha$ towards one
surface. It is reflectively split in point A into a range of diffraction
orders determined by the incident beam angle $\alpha$, the periodic surface
structure described by the lattice spacing $a_{\rm S}$, and the beam
wavelength $\lambda$ through the well known reciprocal lattice equation [49].
We pick two orders, the first one with the reflection angle $\beta$

$\sin\beta=\sin\alpha+\frac{n_{1}\lambda}{a_{\rm S}}\,,$ (1)

with an integer $n_{i}\in\mathbb{Z}$ (numerating the diffraction order), and
the second one with reflection angle $\gamma$

$\sin\gamma=\sin\alpha+\frac{n_{1^{\prime}}\lambda}{a_{\rm S}}\,.$ (2)

At point A, the two selected diffraction orders propagate towards points B and
B′, where they are reflected towards point C and recombine. Point B denotes
the reflection point one diffraction order from point A; thus, the
corresponding incidence angle is $\beta$. To satisfy the recombination of the
beam, the reflection angle $\delta$ has to be of a non-zeroth diffraction
order expressed as

$\sin\delta=\sin\beta+\frac{n_{2}\lambda}{a_{\rm
S}}=\sin\alpha+\frac{(n_{1}+n_{2})\lambda}{a_{\rm S}}\,.$ (3)

Analogously, the reflection at point B′ can be determined by

$\sin\varepsilon=\sin\gamma+\frac{n_{2^{\prime}}\lambda}{a_{\rm
S}}=\cos\alpha+\frac{\left(n_{1^{\prime}}+n_{2^{\prime}}\right)\lambda}{a_{\rm
S}}\,.$ (4)

To satisfy the recombination of the beam at point C, the diffraction of the
incoming beams need to occur under the same diffraction angle, which can be
described mathematically by the relation

$\sin\zeta=\sin\delta+\frac{n_{3}\lambda}{a_{\rm
S}}=\sin\alpha+\frac{\left(n_{1}+n_{2}+n_{3}\right)\lambda}{a_{\rm S}}\,,$ (5)

and

$\sin\zeta=\sin\varepsilon+\frac{n_{3^{\prime}}\lambda}{a_{\rm
S}}=\sin\alpha+\frac{\left(n_{1^{\prime}}+n_{2^{\prime}}+n_{3^{\prime}}\right)\lambda}{a_{\rm
S}}\,.$ (6)

These equations yield a constrain for the diffraction orders

$n_{3^{\prime}}=n_{1}+n_{2}+n_{3}-n_{1^{\prime}}-n_{2^{\prime}}\,.$ (7)

In addition to this angular dependence, the distance between points A and C
needs to be the same for both paths to satisfy the recombination of the beams.
Figure 1 illustrates this condition: the blue and red beamlines need to
recombine in the same point C. Otherwise, they would be reflected without any
spatial overlap to interfere directly. If they are reflected into parallel
beams from different spots, they will interfere in the far field with a phase
shift proportional to the spatial difference between both points. To achieve
interference also in the optical near-field regime for the entire
interferometer, the condition reads

$\tan\beta+\tan\delta=\tan\gamma+\tan\varepsilon\,.$ (8)

Finally, we sum up six parameters characterising a reflective atom
interferometer which have to satisfy the conditions (7) and (8). The latter
can either be used for determining the incidence angle $\alpha$ or by
rewriting the equation

$\displaystyle\tan\left(c+N_{1}\right)+\tan\left(c+N_{1}+N_{2}\right)-\tan\left(c+N_{1^{\prime}}\right)-\tan\left(c+N_{1^{\prime}}+N_{2^{\prime}}\right)=0\,,$
(9)

with $c=\cos\alpha$ and $N_{i}=n_{i}\lambda/a_{\rm S}$, one finds the
following conditions leading to an $\alpha$-independent solution:

$\displaystyle n_{1}=n_{1^{\prime}}+n_{2^{\prime}}\wedge
n_{1^{\prime}}=n_{1}+n_{2}\,.$ (10)

The interference pattern is due to the phase shift along the different optical
paths ABC and AB′C. The path lengths can be determined via these angles for
the path along point B

$b=s\left(\frac{1}{\cos\beta}+\frac{1}{\cos\delta}\right)\,,$ (11)

and along the point B′

$b^{\prime}=s\left(\frac{1}{\cos\gamma}+\frac{1}{\cos\varepsilon}\right)\,.$
(12)

The interference occurs via the superposition of two waves with the same wave
vector ${\bm{k}}$, but are phase shifted with respect to the respective path
lengths, $b-b^{\prime}$. Hence, the phase shifts between the different paths
are given by

$\varphi=k(b-b^{\prime})\,.$ (13)

It can be observed in Eqs. (11) and (12) that the path lengths are
proportional to the slab separation $s$ and, thus, $s$ should be tuned with
respect to the wave vector to maximise the phase shift between both
interfering beams.

Figure 2: Distribution of the diffraction orders on the screen $\varphi$
depending on the incidence angle $\alpha$ for a monolithic atom interferometer
built of silicon with hydrogen passivised surfaces, which are separated by 5
mm and have an extension of 50 mm. The considered wavelength was
$\lambda=0.55\,\text{\AA}$. The purple area describes the dark regions where
no particle will appear. For each incidence angle $\alpha$, the maximum
population of the diffraction order is marked in yellow. The remaining peak
intensities are plotted relative to the maximum intensity according to the
colour scale.

Figure 2 illustrates the positions of the different diffraction for different
incidence angles $\alpha$ for a particular interferometer configuration. It
can be seen that the diffraction orders are strongly separated. All lines are
discontinued due to the finite length of the interferometer, which leads to
some beams escaping the interferometer. These particles will likely hit the
surface and fall into the interferometer; thus, they will not affect the
interference patterns.

Figure 3: Optical paths in a monolithic reflective atom interferometer: a slab
cut into a monolithic crystal of length $d$ (50 mm) and width $s$ (5 mm). A
Helium beam with an incidence angle of $83$ deg (dark blue line) enters the
interferometer. It is diffracted at the hydrogen passivised surfaces with a
lattice constant $a_{\rm S}=3.383\,\rm{\AA}$. The diffracted orders are
reflected two more times until they leave the interferometer. It can be seen
that the third-order ($-3=n_{1}+n_{2}+n_{3}$) diffraction beam will not show
any interference (blue dashed line at $30.32$ deg), see table 1, the second-
order beam at $41.87$ deg (orange lines) will not show any interference due to
equal optical path lengths; the diffraction at $56.10$ deg (green lines), the
zeroth order at $83.00$ deg (red lines) will be measured separately in the
near-field regime, whereas they will interfere in the far-field leading to the
interference patterns depicted in Fig. 4.

###  2.2 Reflection coefficients for the different beam paths inside the
interferometer

In the last section, the conditions for interference were obtained. We now
consider the intensity distribution in the interference signal, described via
a reflection function. This reflection function depends on the incidence and
diffraction angle $\vartheta_{1}$ and $\vartheta_{2}$, respectively. We model
the reflected beam via a Gaussian intensity distribution. Consequently, each
diffraction order has a Gaussian profile which we normalise to the real-valued
probability of each diffraction order $\rho_{n}$

$r(\vartheta_{1},\vartheta_{2})=\sum_{n}\rho_{n}{\rm{e}}^{-\frac{\left(\vartheta_{2}-\theta_{n}\right)^{2}}{2\sigma_{n}^{2}}}\,,$
(14)

with the width of the diffracted signal $\sigma_{n}$ and the position of the
diffracted beam $\theta_{n}$ determined by Eq. (1). The widths depend on the
incidence angle and wavelength,
$\sigma_{n}=\sigma_{n}(\lambda,\vartheta_{1})$. These impacts are negligible
for surface diffraction, the manuscript’s content, due to the overall weak
reflection signal [43]. The reflection coefficient (14) only includes the
inelastic scattering, that the wavelength of the outgoing wave is the same as
that of the incoming wave, $\lambda_{\rm inc}=\lambda_{\rm out}$. Thus, the
total reflected signal is smaller than one,
$\int\mathrm{d}\vartheta_{2}\,r(\vartheta_{1},\vartheta_{2})<1$.

In general, there are five lengths involved in such an interferometer: the
wavelength ($\lambda_{\rm dB}$), the dimensions of the interferometer (length
$d$ and slap separation $s$) and the free-space propagation lengths (source to
interferometer $L_{1}$ and interferometer to detector $L_{2}$). Typically,
these dimensions are on different length scales $\lambda_{\rm dB}\ll
d,s<L_{1},L_{2}$. This consideration allows for the separation of length
scales. Consequently, each particle will only interfere with itself inside the
same optical path in the interferometer. Thus, we can treat each path inside
the interferometer separately, and the collected diffraction image will follow
from the Gaussian beam envelope. The partial waves will experience a different
phase shift due to the optical path (13). Thus, we can describe the reflection
properties of the entire interferometer with a single modified reflection
coefficient

$\displaystyle r_{\rm
inter}(\vartheta_{1},\vartheta_{2})=\sum_{n_{1}n_{2}n_{3}}\rho_{n_{1}}\rho_{n_{2}}\rho_{n_{3}}f_{n_{1}n_{2}n_{3}}{\rm{e}}^{{\rm{i}}kb_{n_{1}n_{2}}}{\rm{e}}^{-\frac{\left[\vartheta_{2}-\theta_{n_{1}n_{2}n_{3}}(\vartheta_{1})\right]^{2}}{2\sigma^{2}}}\,,$
(15)

with the wave vector of the matter wave, $k=2\pi/\lambda$, and the indicator
function $f_{n_{1}n_{2}n_{3}}$ factoring in the interferometer’s geometry
(which determines whether the beam can pass through the interferometer or
not). The beam spread of all diffraction orders will usually be the same for a
monochromatic wave, $\sigma=\sigma_{n}$ for all diffraction orders $n$. Due to
the tilted reflective surfaces with respect to the beam incidence, the
detected spots will be slightly asymmetric, which we neglect for consideration
in this manuscript. The position of the diffraction order is given by
$\theta_{n_{1}n_{2}n_{3}}(\vartheta_{1})$ which is the three-times composition
of Eq. (1) simplifying to

$\theta_{n_{1}n_{2}n_{3}}(\vartheta_{1})=\arcsin\left[\sin\vartheta_{1}+\frac{\left(n_{1}+n_{2}+n_{3}\right)\lambda}{a_{\rm
S}}\right]\,.$ (16)

###  2.3 The monolithic interferometer for He and Si(111)-H$(1\times 1)$

Let us consider a helium beam with de-Broglie wavelength $\lambda_{\rm
dB}=0.55$Å and a beam spread of 1 mrad at a distance of 1 m from the
interferometer (propagation length $L_{1}=1\,\rm{m}$). This corresponds to a 1
mm beam waist ($w=1\,\rm{mm}$) as the beam enters the interferometer. The
lattice spacing of Si(111)-H(1$\times$1) is $a_{\rm S}=3.383$ Å [42]. We
consider an incidence angle of 83 deg and the reflection coefficient (14) with
the amplitudes $\rho_{0}=0.06$, $\rho_{\pm 1}=0.03$ and $\rho_{\pm 2}=0.015$.
These scattering values correspond to the experimentally obtained data for a
beam with a wavelength of $0.6$ Å and an incidence angle of $52$ deg reported
in Ref. [42]. We changed the incidence angle because the result would be
restricted to the zeroth order; see Fig. 2. Table 1 shows the ratio of
transmitted atoms into each diffraction channel. The reflection coefficients
influence only the amplitude of the interference patterns, not the position of
the peaks. Thus, the impact of the correct scattering amplitudes is
neglectable. We have chosen the parameters to demonstrate several effects: the
single-beam transmission, the two-path superposition and the multi-path
interference. Here we restrict our considerations to the zeroth, first and
second diffraction orders. Furthermore, we consider the reflecting plates to
be 50 mm long and 5 mm separated from each other. The optical paths for this
scenario are depicted in Fig. 3. It can be seen that the third-order
diffraction beam (at $30.32$ deg, blue line) consists of a single beam. It
thus will not show any interference; the second-order beam (at $41.87$ deg,
orange line) is the superposition of two paths, as described in Sec. 2.1, but
with equal optical paths which again will not interfere; and the first and
zeroth order will show two separate signals each, that will lead to
interference in the far field.

Angle $\varphi\,(\rm{deg})$ | $n_{1}$ | $\beta(n_{1})$ | $n_{2}$ | $n_{3}$ | path $b\,(\rm{cm})$ | Ampl. $a$ (%)
---|---|---|---|---|---|---
30.32 | 0 | 1.4486 | -1 | -2 | 5.00 | 0.0270
41.87 | -1 | 0.9791 | 1 | -2 | 5.00 | 0.0135
41.87 | 0 | 1.4486 | -1 | -1 | 5.00 | 0.0540
56.10 | -2 | 0.7307 | 2 | -1 | 4.77 | 0.0068
56.10 | 0 | 1.4486 | -2 | 1 | 4.77 | 0.0270
56.10 | -1 | 0.9791 | 1 | -1 | 5.00 | 0.0270
56.10 | 0 | 1.4486 | -1 | 0 | 5.00 | 0.1080
83.00 | -2 | 0.7307 | 1 | 1 | 1.57 | 0.0135
83.00 | -2 | 0.7307 | 2 | 0 | 4.77 | 0.0135
83.00 | 0 | 1.4486 | -2 | 2 | 4.77 | 0.0135
83.00 | -1 | 0.9791 | 0 | 1 | 1.79 | 0.0540
83.00 | -1 | 0.9791 | 1 | 0 | 5.00 | 0.0540
83.00 | 0 | 1.4486 | -1 | 1 | 5.00 | 0.0540
83.00 | -1 | 0.9791 | -1 | 2 | 1.57 | 0.0135
Table 1: Overview of beams expected from an 83 deg incidence angle reflected
into the diffraction angle $\varphi$ with the diffraction orders $n_{1}$,
$n_{2}$ and $n_{3}$ with the first diffraction angle $\beta$ in radians. The
last columns describe the optical path length $b$ and amplitudes $a$ being the
ratio of transmitted atoms into each diffraction channel (%). Figure 4: Far-
field diffraction patterns of each spot at 30.32 deg (blue line), 41.87 deg
(orange line), 56.10 deg (green line) and 83.00 deg (red line) for a Helium
beam with wavelength $\lambda_{\rm dB}=0.55\,\rm{\AA}$ with an incidence angle
of 83 deg. The vertical black dashed line marks the regime for the considered
scenario.

Due to the separation of the length scales and the fact that the atoms
interfere with themselves and not with each other, we describe each
interference pattern via a phase-shifted Gaussian wave in analogy to the
Michelson interferometer. Thus, the interference pattern is described by the
superposition of phase-shifted Gaussian waves

$\displaystyle
I(\varphi)\propto\left|\sum_{n}a_{n}{\rm{e}}^{{\rm{i}}\frac{k\sin\varphi}{2}b_{n}}\right|^{2}{\rm{e}}^{-\frac{2L_{2}^{2}\sin^{2}\varphi}{w^{2}}}\,,$
(17)

with the amplitudes $a_{n}=\rho_{n_{1}}\rho_{n_{2}}\rho_{n_{3}}$ and the
optical path lengths $b_{n}$, which are given in table 1. The widths of the
diffraction orders $\sigma$ are small compared to the width of the Gaussian
envelope, $L_{2}\sin\sigma\ll w$, and, hence, can be neglected. It can be seen
in Eq. (17) that the interference fringes are determined by the wave vector,
$k=2\pi/\lambda_{\rm dB}$. Thus, increasing the wavelength, either by
increasing the particle’s mass or velocity, will reduce the spacing between
the interference fringes. The resulting interference patterns are plotted in
Fig. 4. One can see that the diffraction at $30.32$ deg and $41.87$ deg will
not show any interference features due to the equal optical path lengths of
both optical paths. The remaining two spots will show interference effects
with a contrast of 48.5% for the spot at 56.10 deg and 84.1% for the spot at
83.00 deg. The transmission rates of all channels can be found in table 1:
0.027% of the atoms will be diffracted under the angle of $30.32$ deg, 0.0675%
under $41.87$ deg, 0.1688% under $56.10$ deg, 0.216% under $83$ deg. The
remaining particles will not leave the interferometer. The intensity of a
typical helium beam is so big [50] that a signal fraction of $10^{-4}$ can
easily be detected. The velocity spread will be the limiting quantity to
measure the interference patterns for the helium atom interferometry
configuration depicted in Fig. 4. The velocity spread of a supersonic helium
beam depends on the beam temperature, the nozzle diameter and the reservoir
pressure. This has been treated extensively in the literature; see, for
example, Ref. [51]. A velocity spread causes two different effects: (i) a
broadening of the interference fringes and (ii) a spatial movement of the
entire interference pattern, as illustrated in Fig. 5. Finally, to observe
interference, the velocity spread has to be sufficiently small to not cause a
washing out of the interference fringes. Figure 5 illustrates the positions of
the diffraction order for different wavelengths of the incoming beam with a
fixed incidence angle of $83$ deg. It can be observed that the zeroth order
will stay constant. The remaining orders strongly spread out with increasing
wavelength. As in Fig. 2, the lines are not continuous due to the finite size
of the interferometer. It can be seen in table 1 that the interferometer
splits the wave package at the first diffraction point over $\approx
0.71\,\rm{rad}$.

###  2.4 Quantum reflection interferometer

Quantum reflection occurs on the attractive (outer) part of the atom-surface
interaction potential [52] in contrast to surface scattering, where the
reflection occurs on the repulsive (inner) part of the interaction potential
[53, 54]. It is called quantum reflection because, classically, reflection
cannot occur with an attractive force interaction potential. Quantum
reflection has the very big advantage that an extensive range of atoms and
small, few-atomic molecules that would stick under surface diffraction
conditions display quantum reflection. The disadvantage is that quantum
reflection requires small perpendicular wave vectors. This means that, for a
given wavelength, the spatial extension of a reflective interferometer must be
larger than in the surface scattering configuration for the separated beams to
recombine.

Quantum reflection is less sensitive than surface scattering to defects and
surface contamination because it occurs at larger distances from the surface
[55]. Very large specular reflection coefficients of the order of 50% [56] up
to 90% [57] have been measured. Diffraction via quantum reflection was
recently demonstrated experimentally [58] using helium dimers and trimers with
periodically striped surfaces with micron-sized structures. The paper includes
a comparison of the experimental result with scattering theory based on the
diffraction angle distribution (1), reported in Ref. [59]. There is reasonable
agreement between theory and experiment.

Figure 5: Positions of the diffraction orders [$0^{\rm th}$ (blue); $1^{\rm
st}$ (orange); $2^{\rm nd}$ (yellow); $3^{\rm rd}$ (purple); $4^{\rm th}$
(green); $5^{\rm th}$ (light blue); $6^{\rm th}$ (dark red)] for an incidence
angle of $83$ deg depending on the wavelength of the helium beam.

##  3 Conclusions and Future Work

This paper presents the first proposal for a reflective interferometer for
atoms and molecules. We present calculations for a monolithic configuration
based on experimental scattering results for a room-temperature helium beam
from Si(111)-H(1$\times 1$), showing that a beam splitting of more than 0.5
radians is achievable. Furthermore, we argue that quantum reflection
diffraction is a viable option for extending the beams and surfaces that can
be used and potentially increase the signal intensity. The interference of
larger and complex molecules can be achieved by using different interaction
potentials, such as evanescent fields [53]. A reflective atom or molecule
interferometer, particularly in a monolithic configuration, opens several
possibilities for applications, for instance, as an accelerometer, in
investigating the coherence of matters near dielectric surfaces, as a
continuous velocity selector etc. The next obvious first step is to do a
demonstration experiment of the new interferometer with a helium beam and to
do detailed designs of quantum reflection setups. The latter will require the
calculation of quantum (diffraction) reflection coefficients for a range of
realistic system configurations.

## Acknowledgments

J.F. gratefully acknowledges support from the European Union (H2020-MSCA-
IF-2020, grant number: 101031712).

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