# Analytical Engines With Context-Rich Processing:
Towards Efficient Next-Generation Analytics

Viktor Sanca  EPFL
Lausanne, Switzerland
<EMAIL_ADDRESS>Anastasia Ailamaki  EPFL
Lausanne, Switzerland
<EMAIL_ADDRESS>
###### Abstract

As modern data pipelines continue to collect, produce, and store a variety of
data formats, extracting and combining value from traditional and context-rich
sources such as strings, text, video, audio, and logs becomes a manual process
where such formats are unsuitable for RDBMS. To tap into the dark data, domain
experts analyze and extract insights and integrate them into the data
repositories. This process can involve out-of-DBMS, ad-hoc analysis, and
processing resulting in ETL, engineering effort, and suboptimal performance.
While AI systems based on ML models can automate the analysis process, they
often further generate context-rich answers. Using multiple sources of truth,
for either training the models or in the form of knowledge bases, further
exacerbates the problem of consolidating the data of interest.

We envision an analytical engine co-optimized with components that enable
context-rich analysis. Firstly, as the data from different sources or
resulting from model answering cannot be cleaned ahead of time, we propose
using online data integration via model-assisted similarity operations.
Secondly, we aim for a holistic pipeline cost- and rule-based optimization
across relational and model-based operators. Thirdly, with increasingly
heterogeneous hardware and equally heterogeneous workloads ranging from
traditional relational analytics to generative model inference, we envision a
system that just-in-time adapts to the complex analytical query requirements.
To solve increasingly complex analytical problems, ML offers attractive
solutions that must be combined with traditional analytical processing and
benefit from decades of database community research to achieve scalability and
performance effortless for the end user.

###### Index Terms:

analytics, machine learning, data integration, query optimization, hardware-
conscious processing

##  I Introduction

One of the main goals of data analytics is to extract insights, value, and
intelligence from the available data sources. Traditionally, in the case of
relational database management systems, the data would be stored in tables
over which users declaratively specify what they would like to extract from
the data, with several layers of abstractions performing various logical and
physical optimizations. This takes the complexity, and the responsibility from
the hands of the user and relieves them from specifying how the data
processing pipeline should execute.

However, we are witnessing and participating first-hand in producing context-
rich data, such as documents, text, images, and videos, through social media
or other means. This trend is growing and expected to continue with the
adoption of metaverse-like technologies [1]. The underlying context, such as
sentiment or interest detection from text or object recognition and
classification from image sources, represents a rich source of value, however
unreachable and therefore called dark to the data processing tools.

While traditional database systems lack the means to analyze such unstructured
data, the explosion of research in machine learning has provided both
algorithms and systems fueled by large amounts of data and the latest
technology to enable super-human performance in various inference and
classification tasks. Tasks previously requiring domain experts and
significant intervention of the human-in-the-loop can take place automatically
using artificial intelligence. Also, models such as GPT-3 [2] can also
represent data sources, generating new data. More broadly, Foundation Models
[3] provide an opportunity to adapt large-scale models to specific tasks.
Overall, machine learning provides us with analytical capabilities to process
and transform the data with human-like context-rich semantics while performing
at super-human performance, concerning the speed of processing and the quality
of the result.

Figure 1: Analytics with traditional and model-driven sources and processing.

We group data and insight sources into two categories in Figure 1. The upper
part depicts the sources whose analysis is ML-driven and often have their
software systems run individually. Similarly, we consider a traditional data
source structured data, collected from various sources (sensors, transactions,
domain experts) and cleaned to a certain extent. In a complex data processing
pipeline that needs to combine two sources, the user is faced with complex
decisions, departing from declaratively specifying only what they would like
to achieve. Such manual processing is tedious, error-prone, or inefficient and
would require a high level of system- and programming- in addition to domain-
expertise. To further exacerbate the issue, manually tuning the systems for
the particular query to run on increasingly heterogeneous hardware is
difficult and impractical.

The challenge for performing optimizations and combining data from various
sources (either collected or trained on) is that the data cannot be fully
cleaned and unified to an expected schema and values ahead of time.
Furthermore, the source of dirty data is less likely to be a mistake such as
misspelling but a word with the same semantics (synonym, alternative spelling,
alternative forms) - further representing a context-rich online data cleaning
task.

To outline the design goals and integration challenges, opportunities, and
potential solutions to integrate context-rich processing with analytical
engines (the blue-shaded part of Figure 1), in the rest of the paper, we
describe:

  * •

a motivating example in section II,

  * •

methods for processing context-rich data in section III,

  * •

integration of the context-rich data sources expert-free and on-the-fly in
section IV,

  * •

exposing logical and physical optimizations in section V,

  * •

the challenge of optimizing for hardware heterogeneity in section VI.

We envision an analytical engine that declaratively combines context-rich
processing with traditional data sources to hide from the user the complexity
of logical and physical optimization, underlying hardware, and resulting on-
the-fly data integration.

##  II Motivation

Figure 2: A query that combines both context-rich and traditional analytics.

Suppose an online shopping platform wants to analyze the data and trends using
several sources:

  1. 1.

traditional RDBMS where product, user, and transaction details are managed and
stored,

  2. 2.

a general knowledge base to supplement and extend the product information
based on domain expertise,

  3. 3.

image storage of the products (from reviews, other websites, or social media).

The data analyst wants to answer which clothing products with a price greater
than 20 appear in customer images taken after a specific date, but such that
it is not only the picture of the clothing apparel but other objects appear
too - possibly indicating that it is not a generic product description image.
This query is conceptually represented in Figure 2.

There are three distinct tasks that a data analyst has to perform:

  1. 1.

extract the qualifying data from RDBMS,

  2. 2.

infer which ones are clothing items and combine the results with matching
clothing objects in images,

  3. 3.

analyze the photos taken after a specific date if they contain more than two
objects.

With an RDBMS, step 1) is entirely declarative, and the user can retrieve and
export the qualifying records efficiently. However, if not careful, the user
may not push down the filter in the knowledge base to reduce the qualifying
records for the first join, where only clothing items should be selected.
Furthermore, the knowledge base may have been curated and collected on a
different and broader dataset that does not precisely match the labels or
available data in the RDBMS. These differences are more likely to occur due to
synonyms or some other alternative word or phrase forms that may not be
resolvable with traditional string similarity algorithms and where the user
may need to manually or with the help of an expert resolve which items are
indeed clothing items, which can be both error-prone and heavily involve human
in the loop which increases the effort and execution time. Finally, the
analyst may make a mistake, and instead of first filtering out images after a
specific date, perform heavy processing on all the corpora and similarly have
issues merging the object labels with ones coming from steps 1) and 2).

Overall, the analyst now has the task to carefully orchestrate possibly an ad-
hoc query and multiple sources of data and system components. The cost of
logical or physical optimization mistakes, as well as not considering
scalability and potentially not processing data beyond main-memory capacity,
can quickly hinder even a reasonably simple context-rich query and make it
frustrating to use and design such pipelines, abandoning the potential value
of the data. Furthermore, every query becomes a hand-tuned orchestration
between the components, including logical optimization steps, physical
optimization, and hardware resource management, despite all the lessons
learned from decades of data management research.

Several requirements surface from the example, reiterating the need for
declarative processing where users do not need to specify how, but only what
they want to process. To have a declarative context-rich analysis system, we
propose a design that necessitates the following:

  1. 1.

cost- and rule-based optimization,

  2. 2.

hands-free hardware resource management,

  3. 3.

ad-hoc data matching and integration,

  4. 4.

exposing automated context-rich processing methods.

The broad database and machine learning research community have focused
actively on solving some or all of these issues. In the next section we will
discuss the need for automated context-rich data analysis methods.

##  III Analyzing Context-Rich Data

We broadly use the term context-rich data for all the data sources that would
require a human or a domain expert to infer the meaning (context) that is not
immediately stated as machine-understandable rules. For example, string edit
distance or locality-sensitive hashing-based string similarity can compare
strictly specified characteristics, but such methods cannot capture string
synonyms in a straightforward way, where one would use a dictionary or a
human-in-the-loop. Similar can apply to images, where finding the most
prevalent colors is easy to specify and compute; however, understanding the
image setting and performing object detection and classification would be
considered context-rich analysis.

One way to perform the such large-scale analysis is to deploy crowdsourcing
approaches such as Amazon Mechanical Turk[4] or reCAPTCHA[5], or employ domain
experts. These approaches have proven invaluable in annotating data and create
sources that were, for example, immensely useful in training machine learning
models. However, the human-based analysis would be error-prone, slow, and
expensive for analytics.

On the other hand, machine learning had its renaissance due to the data volume
explosion, hardware processing power improvement, and novel systems and
algorithms, which enabled creating of tools that outperform humans even at
context-rich processing tasks. Tasks such as text, image, audio, and video
analysis are ongoing topics of research, and specific tasks yield even super-
human performance. More importantly, obtaining high-quality models trained on
Web-scale data as a commodity resource that can be modified via mechanisms
such as one- or few-shot learning is possible, for example, via the idea and
initiative of Foundation Models [3]. Context-rich processing must happen
seamlessly and scale with the data and request volume, and model-based
approaches allow high-recall analysis on various tasks.

It is desirable to use models pre-trained on web-scale data with significant
compute resources and fine-tune them to the particular task or dataset at
hand. Creating and using models and running them efficiently is possible
through frameworks such as Tensorflow [6] or PyTorch [7], with further
extensions such as Keras [8] to further make models more declarative.

There has been a significant interest in bringing together statistical
processing with ML training and inference into the traditional data management
systems, using systems such as Raven [9], MADLib [10], or SystemML [11] to
perform inference or learning on tasks such as regression, classification,
clustering to extract richer insights from existing, often tabular data. The
lessons and the need are the same: we need viable integration for context-rich
analysis and logical and physical optimizations for new data sources and value
to avoid the pitfalls of manually constructing complex unoptimized data
processing pipelines.

Since the models have been trained on large-scale data, and conversely, we can
use data to enrich the analytical pipeline from various sources such as
knowledge bases, we cannot assume that data is integrated and cleaned even in
the outputs. Cleaning the data ahead of time is impractical, if not
impossible, due to the scale. Furthermore, models are probabilistic and do not
produce only labels (which can be context-rich on their own), for example, as
in Table I, but generative models can produce output and data on their own
[2]. Integrating the data between the analytical pipelines becomes a key
challenge since online result consolidation typically remains the only option
[12].

TABLE I: Example of context-rich text labels that models may output category | semantic matches
---|---
dog | dog, canine, golden retriever, puppy
cat | cat, maine coon, feline, kitten
animal | cat, dog, golden retriever, feline
shoes | boots, sneakers, oxfords, lace-ups
jacket | blazer, coat, parka, windbreaker
clothes | boots, parka, windbreaker, coat

However, learned approaches allow us to observe this as a representational
learning problem that would allow automated resolution, grouping, joining, and
filtering based on context-similarity (such as synonyms and similar
categories). Approaches such as word2vec and[13], fastText [14] create a
latent representation of strings to learn word associations from large corpora
of text, such as Wikipedia [15] or Common Crawl [16]. Such approaches can
match similarities based on synonyms, semantics, alternative spellings,
different tenses, and even misspellings [17]. Context-based filtering,
joining, and grouping based on similarity is then converted to finding
similarity between original data representation in the latent, high-
dimensional vector space - a task that can be specified and automated.

In the next section, we will discuss how using models to analyze the data and
combining the output from different parts of the query can be expressed as
operators amenable to logical and physical optimizations across different data
processing pipelines.

##  IV Query Processing And Data Integration

Figure 3: Automated, on-the-fly result consolidation.

To enable a variety of logical and physical optimizations of the holistic
query pipeline, no matter where standard relational or model-based operators
are placed, we need to start from a common intermediate representation (IR).
Approaches such as Weld [18] or Raven [9] already take significant steps
towards a common representation that is amenable to optimization rules.

Since we are to use external models in the pipeline, we need to express some
properties of context-rich analysis operators. We need to specify the expected
input and output schema at the logical level. Furthermore, we need to select
the transformations such operator applies on the input, in terms equivalent to
the selection, projection, union, or joins, and what effect operator push-
down/pull-up has on other, for example, relational operators. Finally, in this
step, we must include high-level cost information, such as the effect on the
input/output cardinality. This ensures semantic equivalence and correctness
and can be expressed in existing query optimization frameworks [19].

Next, we consider some physical optimizations related to context-rich
operators using models. While some models perform computationally-intensive
inference only, there are still models that perform similarity search or other
methods which impose different costs based on data access patterns that we
need to enumerate. For example, index-based access for similarity search [20]
should be accounted for in the optimization process in a cost-based
optimization process.

Finally, we must process and optimize by considering the potential polystore
situation where we will have multiple sources of data and multiple processing
engines. To add to this complexity, we also have increasingly heterogeneous
hardware at our disposal (CPU, GPU, TPU, FPGA, ASICs, DPU, etc.), all with
corresponding data transfer, processing, and startup costs. We need to
annotate the processing unit or operator traits and perform decisions that
adapt to the underlying hardware, with principles described in the thesis of
Chrysogelos on efficient analytical processing for CPU-GPU platforms [21].
Just-in-time decisions are needed to make correct decisions in growing
hardware, operator, and system heterogeneity. To fully use the underlying
hardware capabilities, this merits optimizing novel analytical operators
individually for existing or new platforms and holistically for hybrid
execution over a variety of compute units.

Irreplaceable components of context-rich analysis are context-rich semantic
operators that can process, transform, and integrate data with traditional
analytical operators. This task resembles data cleaning but has to capture
context and semantics, perform the task efficiently at query time, and run
without user supervision in an automated way. In particular, consider the
example in Table I: if we want to join clothing items, without strong domain
expertise or knowledge-base, we would not be able to perform matching, and
still, data sources may contain synonyms, spelling differences, or other
expert-system-detectable differences. To resolve this issue, we propose model-
assisted context-rich processing that we call semantic join, which we denote
with a small robot in Figure 2.

In practice, we can use representational models such as fastText [14] to
capture the context even in place of spelling mistakes [17]. More complex
models have the capacity to specialize in performing this task with more
complex input, such as sentences [22, 2], and work with more complex data,
such as images or audio. In particular, model-assisted operators transform the
input to latent, high-dimensional vector space, where distance-based metrics
such as cosine similarity capture features such as context similarity [13]. We
represent this in Figure 3, where a tedious and domain-expert task becomes
completely automated, allowing on-the-fly result consolidation based on
context.

Traditional techniques such as locality-sensitive hashing or other distance
metrics capture syntactic similarity (such as misspellings); however,
previously mentioned deep-learning methods learn the context from large data
corpora. However, other optimizations known to speed up similarity operations,
such as joins, are still applicable and should be explored in this context,
where initial embeddings are already provided instead of created based on
transformations such as hashing.

We propose the following operator extensions:

  * •

Semantic Select \- context-based filtering, e.g.
word = "Clothes" using model "M" with cosine threshold >= 0.9,

  * •

Semantic Join \- joining two or more relations based on join key context, for
example based on model latent space distance between the join key mappings,

  * •

Semantic GroupBy \- on-the-fly clustering of the result based on a model-based
similarity threshold.

With semantic operators, more complex optimization techniques that work for
relational data, such as data-induced predicates, can be evaluated and applied
in the query plans [23]. The key idea is to make the specified unified query
plan amenable to traditional relational plan optimizations.

We note that SQL may not be the best or the only way to represent such query
plans; however having an optimizable representation is required to
systematically perform a variety of transformations that preserve the
correctness of the execution. Systems such as Weld[18], or effort to push
Pandas down the DBMS [24] as a common data science tool - should be considered
in such a next-generation system to keep it familiar and interoperable with
existing libraries and code-ecosystem.

##  V Query Optimization

Figure 4: Additive effects of logical and physical optimizations (log-scale).

Cost- and rule-based optimizations must be considered to produce a scalable,
efficient, and hardware-conscious data-analytics pipeline. Furthermore,
selecting the correct physical operator implementation and scaling the
solution to available compute resources improves hardware utilization and end-
to-end execution time.

We briefly analyze a model-assisted semantic similarity join to demonstrate
the importance of optimizations. To simulate the operation, we join two arrays
of 10k strings taken randomly from the Wikipedia dataset. We use fastText word
embeddings with a dimension of 100, perform the similarity join with cosine
distance where the threshold has to be greater than 0.9, and run the
experiment on a server with two socket Intel Xeon CPU Gold 5118 CPU (2 x
12-core, 48 threads) and 384GB of RAM. We summarize the execution time and the
effects of optimizations in Figure 4.

Suppose the data analyst takes the first tool at their disposal and uses
Python: they load the data eagerly (that may not fit in memory ordinarily) and
then specify the task using the available libraries to iterate over two loops
and perform a similarity check. Suppose this is a part of a more complex
pipeline, and the user did not apply the filter before processing - a standard
filter pushdown rule. This represents the left-most bar, and the variant with
1% selectivity applied beforehand in the blue patterned bar. This optimization
already improves orders of magnitude even in Python in a single-threaded
execution context. Since fastText produces a hash table of known words, we can
further try to optimize the amount of data access by prefetching necessary
data, a physical optimization detail that the user may not be aware of. The
next step takes the execution to tighter code, and fewer function calls
between libraries using C++, tightening the similarity code using CPU-specific
instructions, and scaling up the execution. The effect of well-known
optimizations in the database community brings many orders of magnitude
improvements from the initial manual, intuitive solution. Conversely, applying
such optimizations is tedious and error-prone if done manually; however, the
frustration when waiting for an answer that takes thousands of seconds which
can take less than a second may cause the user to abandon such analysis.

Inefficiencies can quickly add up in a more complex plan, as does the
complexity of manually optimizing and tuning ad-hoc context-rich queries.
Intuitively, performing expensive model inference or processing benefits
equally, if not more, from correct join orders and filter pushdowns that
potentially reduce the input cardinality.

Exposing all the operators to optimizations and how they affect cardinality
and the input/output characteristics is a necessary prerequisite for the
optimization. Frameworks such as Apache Calcite [19] already support flexible
operator characteristics, traits, and transformations. We could encapsulate
such operators in a UDF-like manner while exposing details such as compute
requirements, amenability to parallelizing the input, and memory and data
transfer requirements to the optimizer component. Often, there are already
highly optimized runtimes to perform inference [6] that can reduce development
time and be fused more tightly in global processing orchestration. However, we
have to be mindful that some models that are useful for context-rich analysis,
such as word embedding models, have both inference modules and data structures
such as hash tables and index structures [20] for expediting operations such
as similarity or top-k searches, which have to be included in the optimization
process equally as relational data indexes are.

The key to optimizing a complex context-rich query, such as one in Figure 2,
is to provide correct abstractions that perform the optimizations while
enabling the user to declaratively and in an intuitive way specify the data
processing task at hand, possibly even making well-educated or learned
decisions on some of the semantic operators without explicit user control.

##  VI Hardware-Conscious Optimizations

Machine learning workloads were one of the primary drivers of mainstream GPU
adoption, which was consequently explored in the context of data management on
heterogeneous hardware. Increasingly heterogeneous hardware, with various
computational, data movement, and storage access overheads, is a source of
active research. The general problem becomes how to efficiently scale up the
individual algorithms, tune them to hardware capabilities, orchestrate hybrid
execution models, and schedule many machine-learning inference runtimes on the
available hardware.

We represent the conceptual heterogeneous hardware layout in Figure 5: it
includes multi-core CPUs on multiple sockets, potentially each with a
separately interconnected GPU. Furthermore, a TPU-like device is designed to
perform fast and efficient inference [25], however, that can equally be
considered as an accelerator for relational query processing [26]. It is worth
noting that specialized hardware units for processing real-time neural network
inference are becoming commonplace even in mobile phones, such as Apple Neural
Engine or similar [27]. Finally, there are fast network (e.g., InfiniBand) and
storage (NVMe) solutions, all interconnected with PCIe or other technologies.
The big challenge is how to provision these resources correctly, how to place,
split, and schedule the execution such that the increasingly complex
processing pipeline efficiently uses the available resources, and for example,
which lessons from hybrid CPU-GPU processing [21] we can take and further
generalize.

Figure 5: The increasing hardware heterogeneity is challenging to optimize
for.

With increasingly difficult cost and cardinality estimation, fast sampling
running on modern hardware [28] or speculation techniques [29] can come in
handy to provide mechanisms for practical and adaptive query optimization and
execution.

Late binding to the query requirements concerning hardware and data layout
requirements has become a standard in modern analytics. Just-in-time code
generation using frameworks such as LLVM enables specializing the code paths
and equally adapting to an increasing variety of data input and output
formats. This is increasingly important, as we now consider not only typical
relational formats but repositories that contain context-rich data such as
strings, text, audio, images, and video, where the NoDB paradigm [30] and
adapting to data format heterogeneity at runtime [31] play a crucial role in
handling lazy data access to a variety of data formats. Just-in-time code
generation allows this to be specified as late as query runtime: for example,
if model operator output specification is not known ahead of time, this can be
detected, and the code path generated after the model outputs first data -
with corresponding overheads and delay.

Data layouts, data size, compression effects, and access method selection
become a factor to consider. For example, recent research has yielded
specialized databases for vector processing [32]. Integrating complex
pipelines through a well-coordinated polystore mechanism or a fully generated
pipeline should take different data formats as a first-class problem. For
example, the effects of cache hierarchy or hardware-specific vectorized
instructions can significantly change the behavior of processing a high-
dimensional vector. Furthermore, optimization opportunities such as inference
using hardware-enabled half-precision (or lower) floating point formats need
to be considered and specialized for [33].

Finally, data movement and compute shipping have to be carefully designed.
Obvious questions such as data compression before sending the data over the
interconnect for processing come to mind. However, we should also consider the
cost of shipping and initializing model parameters and runtime on devices as a
potential cost and optimization, as complex models can have many millions of
parameters [3]. The decision of the concrete operator execution happens in a
managed runtime such as Tensorflow [6] or being optimized entirely using a
relational pipeline [9], further opening up the landscape of scheduling and
optimizing complex data-science pipelines to use available heterogeneous
hardware resources adaptively based on query-specific requirements.

##  VII Related Work

The tasks analytical engines are supposed to support and optimize are evolving
with novel algorithms, sources of values, applications, and hardware
platforms. The research community and industry are proposing systems and
investing significant research efforts to address them. We can use them as
foundation blocks for next-generation analytical engines for context-rich
processing.

###  VII-A Querying and Optimizing Complex Analytical Pipelines

To extract deeper insights from the data, more complex data transformations
using User Defined Functions (UDFs) or external libraries can take place. In
our case, context-rich analysis can happen as a UDF or by invoking another
framework optimized to perform specific tasks (e.g., image object detection).
A mounting challenge is optimizing the external operators in the query, where
systems like Froid [34] optimize UDFs for Relational DBMS, Weld [18] reduces
the cost of data movement through optimizing a standard IR, Raven [9]
optimizes ML inference with an RDBMS, while Caesar [35] aims to minimize the
overheads between application boundaries. MADLib [10] or SystemML [11] allow
training and inference over the input, often tabular data. Frameworks such as
TensorFlow perform dataflow optimizations for model training and inference
[36]. We envision a system not limited to using models representing only the
data in RDBMS but can cross-optimize and use the models from different data
sources in a polystore-like design. Our vision aims to combine insights and
value from, for example, relational data, knowledge bases, and images in a
single analytical pipeline and holistically optimize the analytical pipeline
across models and frameworks.

Rather than specifying the queries using languages such as SQL, using a
standard Python data analysis library like Pandas and the work on how to
expose it to RDBMS optimizations [24] could be a good step to improve
usability and reduce the potential complexity of specifying new operators that
would be needed for context-rich analytical engine. The end goal would be
taking the existing code in languages such as Python and optimizing it
automatically by creating a standard IR, detecting physical and logical
optimizations, and correctly tuning the operators to the underlying hardware
capabilities, computation optimizations, and data transfers.

###  VII-B System Heterogeneity and Data Integration

The analytical engine components that can perform context-rich analysis are
part of reality and exist in individual components. Our vision is to bring
them together in an efficient engine that would relieve users from performing
tasks other than declaratively specifying what they would like to analyze in
the data.

Foundation Models [3] offer a way to democratize models that would be
practically impossible to retrain individually due to the required data and
computation scale. However, such models can be specialized for various tasks
to perform better than a human counterpart on context-rich data.
Representational Models such as word2Vec[13] and fastText[14] allow learning
the word context and reasoning about such abstract similarity by embedding the
text and performing similarity operations in latent space.

Since some of the models are already trained on web-scale data, and the
analyst will likely supplement the knowledge with external data sources such
as knowledge bases, performing data cleaning tasks on all the input data is
impractical, if not impossible. However, data cleaning remains crucial in
integrating data from different sources. To this extent, we need to
incorporate relevant data in an online rather than ahead-of-query offline
process through a query-driven mechanism previously explored in the context of
repairing functional dependency violations [12].

###  VII-C Adapting to Data and Hardware Heterogeneity

New applications, including machine learning, drive an increasingly
heterogeneous hardware landscape. From the CPU-dominant landscape, we have a
variety of compute units such as CPU, GPU, TPU, DPU [37], FPGA, ASICs, and
others - not only on high-end servers but also on commodity devices such as
mobile phones, such as Apple Neural Engine [27]. This is supplemented with
various storage devices and interconnects that add to the opportunities for
hybrid execution on multiple hardware components and novel optimizations for
out-of-core algorithms.

To adapt to the hardware heterogeneity and complex requirements, analytical
engines such as Hyper [38], NoisePage [39], and Proteus [40] deploy just-in-
time decisions via code generation using frameworks such as LLVM [41]. On the
side of machine learning, systems such as TensorFlow use JIT-compilation and
specialize the runtime graph utilizing an approach called XLA using a domain-
specific compiler.

There has been a significant effort in developing optimization frameworks,
such as Apache Calcite [19] for performing logical, rule- and cost-based
optimization over heterogeneous data sources. Furthermore, at a lower level of
optimization, frameworks such as MLIR [42] provide a hybrid IR which can aid
in bringing together different system and software components and optimizing
them for heterogeneous hardware under unified infrastructure.

##  VIII Towards Next-Generation Analytics

In addition to the typical structured data analytics, humans produce and
collect data that can contain value locked behind rich context for a
traditional analytical engine. This may further get exacerbated by proposed
use cases such as metaverse, where combining analytical insights from sources
of interaction such as images, text, audio, and traditional data sources
presents an untapped source of insight.

While context-rich analysis, such as object detection for images or sentiment
analysis in text, has proposed solutions using machine learning techniques,
these solutions are often optimized individually. For more complex analytical
pipelines, the user would perform physical, logical, and hardware
optimizations for potentially ad-hoc analytical queries, which we want to
avoid through a unified, next-generation analytical engine that applies the
lessons learned in data management.

With many specialized and siloed solutions, both ML and DB communities make
strides to enable richer and more efficient solutions to extract insights and
value from data. We propose a system that performs holistic data processing
pipeline optimizations merges the potentially unexpected or dirty results on
the fly, and relieves the user from making decisions over the increasingly
complex system and hardware landscape.

We envision a next-generation analytical engine that generalizes analytical
operations driven by new data processing methods to enable novel ways to
extract and combine insights from different sources. At the same time, users
are given control of data integration through a new class of semantic and
model-assisted relational operators that enable automated result
consolidation. Through vertical optimizations, correct abstractions, and just-
in-time adaptation, a single declarative framework aims to reach the
performance of individually and manually tuned components.

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# Reservoir optics with exciton-polariton condensates

Y. Wang  School of Physics and Astronomy, University of Southampton,
Southampton, SO17 1BJ, UK.  H. Sigurdsson<EMAIL_ADDRESS>School of
Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK.
Science Institute, University of Iceland, Dunhagi 3, IS-107, Reykjavik,
Iceland  J. D. Töpfer  School of Physics and Astronomy, University of
Southampton, Southampton, SO17 1BJ, UK.  Skolkovo Institute of Science and
Technology, Moscow, Territory of innovation center “Skolkovo”, Bolshoy
Boulevard 30, bld. 1, 121205, Russia.  P. G. Lagoudakis  Skolkovo Institute of
Science and Technology, Moscow, Territory of innovation center “Skolkovo”,
Bolshoy Boulevard 30, bld. 1, 121205, Russia.  School of Physics and
Astronomy, University of Southampton, Southampton, SO17 1BJ, UK.

###### Abstract

We investigate an all-optical microscale planar lensing technique based on
coherent fluids of semiconductor cavity exciton-polariton condensates. Our
theoretical analysis underpins the potential in using state-of-the-art spatial
light modulation of nonresonant excitation beams to guide and focus polariton
condensates away from their pumping region. The nonresonant excitation profile
generates an excitonic reservoir that blueshifts the polariton mode and
provides gain, which can be spatially tailored into lens shapes at the
microscale to refract condensate waves. We propose several different avenues
in controlling the condensate fluid, and demonstrate formation of highly
enhanced and localised condensates away from the pumped reservoirs. This opens
new perspectives in guiding quantum fluids of light and generating polariton
condensates that are shielded from detrimental reservoir dephasing effects.

##  I Introduction

Advancements in guiding and focusing the flow of planar (paraxial) light waves
at the microscale brings far-reaching possibilities into miniaturized optical
technologies, from microlens arrays [1] to optical circuitry and logic gates
[2], that are reliant on dispersion management. Metamaterials [3, 4],
plasmonic lenses [5, 6, 7], phase-change materials [8], photonic crystals [9,
10] and disordered materials [11] all offer a variety of techniques to focus
planar light, though usually coming at the cost of irreversible fabrication
methods. Here, we introduce an all-optical planar microlensing approach in a
system of microcavity exciton-polariton condensates that offers flexible and
reprogrammable lens configurations.

Exciton-polaritons (from here on polaritons) are boson-like quasiparticles
formed by coherent hybridization of electron-hole pairs in semiconductor
quantum wells and microcavity photons in the strong-coupling regime [12], as
sketched in Fig. 1. The extremely small effective polariton mass ($\sim
10^{-5}$ of the electron mass) and large interaction strength, due to the
excitonic component, has opened up new strategies in all-optical control over
macroscopic coherent matter-wave fluids of light, or polariton condensates
[13]. For the past ten years there have been several important experiments in
all-optical manipulation of polariton condensates using nonresonant excitation
methods [14, 15] such as condensate amplification [16], trapping [17, 18],
exceptional points [19], dissipative annealing of the XY model [20], vortex
manipulation [21, 22], and lattices [23, 24]. Many works have also combined
the optical control provided by nonresonant lasers in conjunction with
engineered photonic potentials such as micropillars, microwires, or wedged
cavities (i.e., photonic potential gradient) which led to development of
optically controllable interferometers [25] and transistor switches [26, 27].
Alongside these developments in optical control, there is a growing variety in
cheaper room-temperature materials that operate in the strong coupling regime
[28, 29, 30] which opens new perspectives on the role of exciton-polaritons in
future optical based technologies [27].

Figure 1: Sketch of the nonresonant lensing effect with exciton-polariton
condensates in a semiconductor microcavity. Quantum well excitons are
photoexcited through nonresonant pumping (dark red profiles) and while the
cavity mirrors (symmetric distributed Bragg reflectors) provide photon
confinement and strong-coupling.

In this study, we explore spatial control over propagating exciton-polariton
condensates using nonresonant excitation beams shaped into a planoconcave
microlens (see Fig. 1). The excitation beam induces (photoexcites) a static
reservoir of incoherent excitons which provide both gain and blueshift to the
polariton modes [14, 15]. Consequently, excited polaritons experience a
complex valued effective potential landscape which can both amplify and phase-
modulate transmitted waves. When the excitation beam is removed the reservoir
rapidly decays which permits rewriting new and different potential landscapes
on the same sample location. Recently, similar flexibility was demonstrated
with phonon-polaritons in hexagonal boron nitride heterostructure [31]. So
far, there have been several studies addressing the potential in nonresonant
all-optical control to manipulate the flow of condensate polaritons [16, 32,
33, 26, 34] but, to our knowledge, there has been no investigation on planar
microlensing. Here, we demonstrate strong focusing of polariton waves outside
of their pumped regions in both the steady state, and a multi-energy-component
state. The latter is characterized by tunable, and high contrast, intensity
beatings at the focal point reaching frequencies as high as $250$ GHz.

The rest of the paper is organized as follows: In Sec. II, as a preparation
for the more complicated non-linear polariton system, we study microlensing in
a damped two-dimensional (2D) Schrödinger system corresponding to the linear
(non-interacting) polariton regime. In Sec. III, we nonresonantly excite a
source condensate which approximately emits a coherent plane wave polariton
flow. This flow impinges on a planoconcave microlens generated by a second
nonresonant beam. We analyse the response of the condensate waves against this
additional microlens potential and develop an argument for the operational
requirements of efficient reservoir lensing. In Sec. IV, we investigate a
simpler idea of using only a lens-shaped beam pumped above condensation
threshold, resulting in spontaneous formation of condensate profiles strongly
focused away from their pumped region. Finally, the general conclusion of our
reservoir optics scheme is drawn in Sec. V.

##  II Planar polariton microlensing in the linear regime

We start our analysis by considering first non-interacting (linear regime)
lower branch polaritons in a planar microcavity described, in the effective
mass approximation, with a 2D Schrödinger equation with a complex potential
and drive term representing a resonant laser excitation,

$\displaystyle i\hbar\frac{\partial\Psi}{\partial t}=$
$\displaystyle\left[-\frac{\hbar^{2}\nabla^{2}}{2m}+(V_{r}+iV_{i})V({\mathbf{r}})-\frac{i\hbar\gamma}{2}\right]\Psi$
$\displaystyle+E(\mathbf{r})e^{-i(\omega_{s}t-\mathbf{k}_{s}\cdot\mathbf{r})}.$
(1)

Here, $m$ is the effective polariton mass, $\gamma$ is the linear decay rate
due to the lossy cavity mirrors, $V_{r}$ and $V_{i}$ quantify the the real and
imaginary parts of the potential whose spatial profile $V(\mathbf{r})$ is
taken to have a step-function boundary for brevity, and $E(\mathbf{r})$ is a
coherent (resonant) driving field with frequency $\omega_{s}$ and wavevector
$\mathbf{k}_{s}$.

We will focus on steady state solutions
$\Psi(\mathbf{r},t)=\psi(\mathbf{r})e^{-i\omega_{s}t}$ giving the time-
independent Schrödinger equation,

$\hbar\omega_{s}\psi=\left[-\frac{\hbar^{2}\nabla^{2}}{2m}+(V_{r}+iV_{i})V({\mathbf{r}})-\frac{i\hbar\gamma}{2}\right]\psi+E(\mathbf{r})e^{i\mathbf{k}_{s}\cdot\mathbf{r}}.$
(2)

Inside the potential $V(\mathbf{r})$ we obtain the homogeneous Helmholtz
equation,

$\displaystyle\nabla^{2}\psi+k^{2}\psi=0,$ (3) $\displaystyle
k^{2}=\dfrac{2m}{\hbar^{2}}\left[\hbar\omega_{s}-V_{r}-i\Big{(}V_{i}-\dfrac{\hbar\gamma}{2}\Big{)}\right].$
(4)

Under resonant driving, $k_{s}=\sqrt{2m\omega_{s}/\hbar}$, the refractive
index of the complex-valued potential with respect to the source is,

$\displaystyle
n^{\prime}=\sqrt{1-\frac{V_{r}}{\hbar\omega_{s}}-\frac{i}{\hbar\omega_{s}}\Big{(}V_{i}-\frac{\hbar\gamma}{2}\Big{)}}.$
(5)

Assuming $V_{r}<\hbar\omega_{s}$ and
$V_{i}-\hbar\gamma/2\ll\hbar\omega_{s}-V_{r}$ we can Taylor expand Eq. (5) so
it reads,

$n^{\prime}=n+i\kappa,$ (6)

where

$n=\sqrt{1-\dfrac{V_{r}}{\hbar\omega_{s}}},\qquad\kappa=-\dfrac{2V_{i}-\hbar\gamma}{4\hbar\omega_{s}n}.$
(7)

For a planoconcave shaped potential $V(\mathbf{r})$ whose edge is depicted
with a green solid line in Figs. 2(b) and 2(e) (see Supplemental Material for
the case of a positive meniscus lens) we recall the Lensmaker’s equation in
the ray optics limit where the focal length $f$ follows,

$\displaystyle f=\frac{R}{1-n}.$ (8)

Here, $R$ is the radius of curvature of the back surfaces of the lens. In the
case of planar microlenses, whose characteristic spatial scale is only several
wavelengts, the focal length will deviate from Eq. (8) due to pronounced
scattering and interference of the waves impinging on the lens. We thus
numerically solve the steady of Eq. (II) under resonant excitation of plane
waves that pass through the planoconcave microlens. We base our parameters on
state of the art inorganic microcavities for generating polariton condensates
[35]: $\gamma^{-1}=5.5\mathrm{\,ps}$ and $m=4.9\times 10^{-5}m_{0}$ where
$m_{0}$ is the free electron mass.

((a))

((b))

((c))

((d))

((e))

((f))

Figure 2: (a,b) The normalized density, $|\Psi|^{2}$, and (d,e) phase,
$\text{arg}{(\Psi)}$, of the wave function without (a,d) and with (b,e) the
lens potential. The source and the lens are outlined with green solid lines.
(c,f) Line profile of the wavefunction density along the lens axis ($y=0$) as
a function of varying real (c) and imaginary (f) part of the potential. In (b)
and (e) the yellow dashed lines indicate the principal plane of the lens and
the focal point with distance $f$. The green vertical dashed lines in (c) and
(f) indicate the source location and the front and back surface of the lens.

Since $n=1$ and $n<1$ outside and inside the lens, respectively, an incident
planar wavefront from the left will transmit through the lens to converge into
a cylindrical wavefront on the right side. We numerically solve for the steady
states of Eq. (II) under resonant excitation at the left side of the lens with
a profile,
$E(\mathbf{r})=\mathrm{exp}[-x^{2}/(2\sigma^{2}_{x})-y^{2}/(2\sigma^{2}_{y})]$
whose full-width-half-maximum (FWHM) is outlined with green in Fig. 2(a)
centered at $x=0$. We set the energy of the source excitation to
$\hbar\omega_{s}=$2.0\text{\,}\mathrm{meV}$$ so to have a rapidly varying
phase front and remaining within the parabolic (dispersion) regime. The
corresponding steady state density and phase profiles of $\Psi(\mathbf{r})$
without any lens potential are shown in Figs. 2(a) and 2(d).

When a planoconcave potential $V(\mathbf{r})$, of size in the order of several
wavelengths ($\lambda_{s}\approx 3.9$ $\mu$m), is introduced both transmitted
and scattered waves contribute in a complicated way to the focal region on the
right side of the lens [see Fig. 2(b)]. Here we set
$R=$10.0\text{\,}\mu\mathrm{m}$$ and the lens’ thickness
$T=$4.5\text{\,}\mu\mathrm{m}$$ which corresponds to $n\approx 0.4472$ and
$f\approx$18.1\text{\,}\mu\mathrm{m}$$ indicated by the yellow dashed lines in
Figs. 2(b) and 2(e). We observe a focal region (i.e., the white-ish region of
converged/focused waves) that lies outside the lens curvature $R$ and within
the ray-optics focal length $f$ [Eq. (8)], as a consequence of the microscopic
nature of the lens shape. We stress that the low polariton intensity in the
focal region is dominated by their rapid decay rate $\gamma$ used in our
simulation. However, condensation of polaritons with large lifetimes reaching
$\gamma^{-1}=270$ ps has also been demonstrated [36], leading to longer
propagation lengths and timescales to manipulate the condensate flow.

In Fig. 2(c), we show the line profile of the wavefunction density at $y=0$
for varying real potential strength $V_{r}$. The focal region both shrinks and
the focal length decreases as the potential strength increases in
qualitatively agreement with Eq. (8) (yellow solid line). It is worth
mentioning that from Eq. (7) one can, in principle, achieve epsilon-near-zero
($n=0$) lensing, which has been studied extensively in metamaterials [37], by
tuning the excitation frequency. However, at $\hbar\omega_{s}\approx V_{r}$
incident waves undergo stronger reflection leading to a pronounced
interference pattern like seen in Fig. 2(b) to the left of the lens. We also
vary the imaginary part of the potential $V_{i}$ in Fig. 2(f) showing a clear
amplification of the transmitted waves in accordance with the imaginary part
of the refractive index in Eq. (7).

##  III Planar reservoir microlensing with polariton condensates

###  III.1 Generalized Gross-Pitaevskii model

Having characterized the effects of the 2D planoconcave microlens on an
incoming plane wave, we now move to the nonlinear regime with condensates of
polaritons. The polariton condensate wavefunction $\Psi(\mathbf{r},t)$ obeys a
generalized Gross-Pitaevskii equation coupled to a driven exciton reservoir
$N(\mathbf{r},t)$ rate equation [38],

$\displaystyle i\hbar\frac{\partial\Psi}{\partial t}$ $\displaystyle=$
$\displaystyle\bigg{[}-\frac{\hbar^{2}\nabla^{2}}{2m}+\alpha|\Psi|^{2}+G\left(N+\frac{\eta
P(\mathbf{r})}{\Gamma}\right)$ (9) $\displaystyle+\frac{i\hbar}{2}\left(\xi
N-\gamma\right)\bigg{]}\Psi,$ $\displaystyle\frac{\partial N}{\partial t}$
$\displaystyle=$
$\displaystyle-\left(\Gamma+\xi|\Psi|^{2}\right)N+P(\mathbf{r}).$ (10)

Here, $G=2g|\chi|^{2}$ and $\alpha=g|\chi|^{4}$ are the polariton-reservoir
and polariton-polariton interaction strengths, respectively, $g$ is the
exciton-exciton dipole interaction strength, $|\chi|^{2}$ is the excitonic
Hopfield fraction of the polariton, $\xi$ is the scattering rate of reservoir
excitons into the condensate, $\Gamma$ is the reservoir decay rate, $\eta$
quantifies additional blueshift coming from a dark background of excitons
which do not scatter into the condensate, and $P(\mathbf{r})$ is the
nonresonant continuous-wave pump. The parameters used in all simulations are
based on negatively detuned cavities, $|\chi|^{2}=0.4$, with GaAs-type quantum
wells, $g=1\,\mu\mathrm{eV\,\mu m^{2}}$. Remaining parameters are taken
similar to those used to describe past experiments, $\hbar\xi=2.8g$; $\eta=5$;
and $\Gamma=\gamma$.

Let us quantify the nonresonant pump as $P(\mathbf{r})=P_{0}f(\mathbf{r})$
where $P_{0}$ is a positive scalar denoting the power density of the pump
laser and $f(\mathbf{r})$ is its profile. It is instructive to define the
condensation threshold which, formally, is a bifurcation point separating the
so-called normal (uncondensed) state ($|\Psi|=0$) and the condensed state
($|\Psi|\neq 0$). The threshold can be identified as the point where a single
frequency component of our system in the linear regime crosses from negative
to positive imaginary value (i.e., small $|\Psi|$ starts growing exponentially
in time). Alternatively, one can also estimate the threshold of Eq. (9)
numerically by expanding the reservoir steady state,

$N=\frac{P(\mathbf{r})}{\Gamma+\xi|\Psi|^{2}}=\frac{P(\mathbf{r})}{\Gamma}\left[1-\frac{\xi|\Psi|^{2}}{\Gamma}+\mathcal{O}{(|\Psi|^{4})}\right]$
(11)

and compare the contribution between the zeroth and the first order terms.
Integrating through space we can write the following inequality:

$\frac{\xi}{\Gamma}\int f(\mathbf{r})|\Psi|^{2}\,d\mathbf{r}<\epsilon.$ (12)

Here, $\epsilon\ll 1$ is some small, reasonably chosen, numerical tolerance to
determine the threshold. Physically, the above expression simply states that
around threshold any nonlinear effects on the reservoir are small. In this
weak nonlinear regime the potential generated by the pump is approximately,

$V(\mathbf{r})\simeq\frac{P(\mathbf{r})}{\Gamma}\left[G\left(1+\eta\right)+i\frac{\hbar}{2}\xi\right].$
(13)

Separating the real and imaginary parts gives,

$V_{r}=\frac{P_{0}}{\Gamma}G(1+\eta),\qquad
V_{i}=\frac{P_{0}\hbar\xi}{2\Gamma}.$ (14)

For a homogeneous pump $P(\mathbf{r})=P_{0}$ the threshold power corresponds
to the balance of gain and dissipation $V_{i}-\hbar\gamma/2=0$ which gives
$P_{0,th}=\gamma\Gamma/\xi$. For inhomogeneous pump spots the threshold power
is bigger due to additional planar losses of waves from the spatially finite
gain region.

The pump which induces the lens potential is constructed using Gaussian
blurring on a step function which naturally mimics the limited resolution of
using spatial light modulators in experiment and finite exciton diffusion,

$f(\mathbf{r})=\int
F(\mathbf{r}^{\prime})e^{-|\mathbf{r}-\mathbf{r}^{\prime}|^{2}/2w^{2}}\,d\mathbf{r}^{\prime}$
(15)

where

$F(\mathbf{r})=\begin{cases}1&\text{for }\mathbf{r}\in\mathcal{L}\\\
0&\text{else}.\end{cases}$ (16)

where $\mathcal{L}$ is the lens area. For all lens shapes used in calculations
of Figs. 3-5 we apply a Gaussian blur corresponding to
$w\approx$0.85\text{\,}\mu\mathrm{m}$$ ($2.0\text{\,}\mu\mathrm{m}$ FWHM).

###  III.2 Numerical results on reservoir lensing

We will consider two separate pumps
$P(\mathbf{r})=P_{S}(\mathbf{r})+P_{L}(\mathbf{r})$, of characteristic sizes
$D_{S}$ and $D_{L}$, which are referred to as the source and the lens as
introduced in Sec. II and depicted in Fig. 1. We will denote the complex-
valued potential coefficients for the source and the lens potentials as
$V_{S}=V_{r,S}+iV_{i,S}$ and $V_{L}=V_{r,L}+iV_{i,L}$, respectively.
Conservation of energy tells us that polaritons generated at the source will
obtain kinetic energy following:

$V_{r,S}=\frac{2\hbar^{2}\pi^{2}}{m\lambda^{2}}.$ (17)

Let us list some requirements in order to obtain steady state lensing of
polariton waves with wavelength $\lambda$ coming from the source and passing
through the lens:

((a))

((b))

((c))

((d))

((e))

((f))

Figure 3: (a) Normalized density $|\Psi|^{2}$ and (d) phase map
$\text{arg}{(\Psi)}$ of the condensate in the steady state under nonresonant
pumping. (b,e) Time-resolved density line profile along $y=0$ for two
different pump powers. (c) Time-integrated density line profile, and (f)
corresponding spectral weight for varying lens power. Data is normalized at
each step in $P_{L}/P_{L,th}$. The FWHM of the pump profiles is outlined with
green solid lines. Vertical green dashed lines indicate the outer and inner
boundary of the source and lens at $y=0$. The horizontal green dashed line
indicates the threshold of the isolated lens. The horizontal yellow dashed
line indicates the onset of periodic dynamics characterized by more than one
spectral peak.

  1. (i)

$D_{L}>\lambda$, the lens has to be large enough to refract the incident
waves.

  2. (ii)

$0<V_{r,S}-V_{r,L}=\Delta$, waves must be propagating in the lens.

  3. (iii)

$P_{L}<P_{L,\text{th}}$, the lens should be below threshold.

  4. (iv)

$P_{S}>P_{S,\text{th}}$, the source must be above threshold.

Here, $P_{S(L),\text{th}}$ are the threshold powers of the isolated source
(lens) pumps.

We can rewrite requirements (i) and (ii) in terms of the model parameters,
respectively,

$\displaystyle
D_{L}>\hbar\pi\sqrt{\frac{2\Gamma}{mP_{0,S}G(1+\eta)}}=\lambda,$ (18)
$\displaystyle 0<\frac{P_{0,S}-P_{0,L}}{\Gamma}G(1+\eta)=\Delta.$ (19)

Here, $P_{0,S(L)}$ denotes the power density of the nonresonant source (lens)
pump. It therefore becomes evident that increasing $P_{0,S}$ will allow us to
satisfy both requirements. However, $\Delta$ needs to be reasonably bounded to
obtain good focusing of transmitted waves. This is evident from the variable
maximum intensity in the focal region in Fig. 2(c). Therefore, arbitrarily
increasing $P_{0,S}$ does not guarantee good focusing of polariton waves. We
also note that requirement (iii) is not strict as we will see later.

We demonstrate our reservoir lensing scheme in Fig. 3 by numerically solving
the generalized Gross-Pitaevskii and reservoir model. We set the profile of
the source pump to be cigar-shaped to approximately generate plane waves
$f_{S}(\mathbf{r})=\mathrm{exp}[-x^{2}/(2\sigma^{2}_{x})-y^{2}/(2\sigma^{2}_{y})]$
in which $\sigma_{x}\ll\sigma_{y}$. The lens is taken to be planoconcave
shaped with $R=$10.0\text{\,}\mu\mathrm{m}$$ and
$T=$4.5\text{\,}\mu\mathrm{m}$$. The FWHM of the source and lens are outlined
with green solid curves in Figs. 3(a) and 3(d). We stress that due to the
different profiles of the source and the lens their threshold powers are
different.

One of the main differences between the resonant scheme discussed in Sec. II
and the current nonresonant scheme is the vivid localization of the source
condensate along the vertical direction shown in Fig. 3(a). This effect stems
from the anisotropic gain region favoring modes with minimal losses, and
effective attractive interactions between the condensate and the reservoir due
to the gain-saturation mechanism [39]. None-the-less, enhancement of
propagating waves in the focal region can be observed clearly in Fig. 3(a),
partly due to amplification from the lens gain. The phase map shown in Fig.
3(d) is very different from that in Fig. 2(d) which stems from the large
detuning between the source waves and the lens potential in simulation, i.e.
$\Delta=V_{r,S}-V_{r,L}\approx 2.0\text{ meV}-0.8\text{ meV}=1.2\text{ meV}$.
In order to reduce the detuning $\Delta$, and get stronger focusing, one could
pump the lens harder. However, this triggers condensation inside the lens and
requirement (iii) is violated. Moreover, reinforcing “ballistic” interactions
between the source and the lens region have lowered the lens threshold [40,
41] (yellow dashed line in Fig. 3). These complex wave dynamics make it
therefore a nontrivial task to adjust the detuning arbitrarily $\Delta$ to
obtain stronger focusing while—at the same time—keeping the lens pump below
threshold.

If, on the other hand, requirement (iii) is relaxed and the lens power is made
variable then interesting nonlinear physics become enhanced. In Fig. 3(c) we
show the time-integrated line profile of the wavefunction density at $y=0$ for
varying lens power $P_{L}$, and the corresponding energy spectrum in Fig.
3(f). As discussed at the start of the section, the system favors a steady
state behaviour when $P_{L}$ is small, characterized by a single clear
spectral line in Fig. 3(f). In this regime, the results are similar to those
of a static lens potential impinged by resonantly excited waves discussed in
Sec. II. However, as the lens power increases, an additional spectral line
appears and nonstationary periodic solutions form as a results of intricate
interactions between the condensate polaritons generated at the source and the
lens, in agreement with recent experiments [42]. An example of two such
solutions in the time domain is shown in Figs. 3(b) and 3(e). Clear $\approx
252$ GHz intensity beatings in the focal region can be observed in Fig. 3(b)
whereas Fig. 3(e) shows two dominant beat frequencies.

##  IV Reservoir lenses above threshold

((a))

((b))

((c))

((d))

Figure 4: (a,b) Steady state condensate density $|\Psi|^{2}$ and (c,d) phase
$\text{arg}{(\Psi)}$ for two different nonresonant pump configurations. The
nonresonant pump is shaped into a (a) planoconcave lens showing clear focusing
of the emitted waves outside the pumping area, and a (b) planoconcave
resonator made from two lenses (emitter) facing each other. Note that each
individual lens is below threshold but the system/resonator as a whole has a
lower threshold and thus supports a standing wave condensate at lower powers.
The FWHM of the pump profiles is outlined in green.

There are limitations to the source and lens scheme in previous section which
cannot be quantified nicely given the complex wave dynamics at play. Firstly,
reinforcing behaviour between the source and the lens regions results in
lowered threshold gain of the interacting system which can lead to
condensation into extended quasinormal standing wave modes that are supported
by both the source and the lens region. This is a general feature of
interacting dissipative systems, such as coupled lasers, or interacting
polariton condensates [40, 41]. Second, the source pump size would, in
general, need to be larger than the lens in order to avoid $\Delta$ getting
too large (i.e., smaller source pumps need to be driven with higher power and
thus emit waves with higher energy). This can lead to thermally induced self-
trapping of the source condensate [43].

To overcome these issues, we consider a more simple case where the source pump
$P_{S}(\mathbf{r})=0$ is omitted and just the lens $P_{L}(\mathbf{r})$ is
driven above threshold. Indeed, the lens region then plays the role of a
carefully designed anisotropic planar emitter from which waves radiate to
constructively interfere. In Fig. 4 we show the condensate steady state for a
pump profile shaped into a planoconcave lens and driven above threshold.
Polariton waves generated in the pump region are propagating along the
direction normal to the lens surface and form a strong focal region with a
clear phase shift.

When the “lens” power is increased then the contrast between the condensate
density within and outside the lens region increases as shown in Figs. 5(a)
and 5(b), where in the latter we plot the condensate density line profile
along the lens axis [44]. These results underpin the potential of using
anisotropic shaped nonresonant excitation beams to generate high density
polariton condensates spatially separated from any influence of the background
exciton reservoir such as strong dephasing or spatial hole burning effects. In
the Supplemental Material we also provide results on a pump shaped into a
positive meniscus lens.

((a))

((b))

Figure 5: (a) Condensate density for a planoconcave shaped nonresonant pump at
high powers and (b) line profile along $y=0$ for varying power. The vertical
green dashed lines indicate the pumped region.

We also investigate the potential of designing planar resonators by setting
two identical lens-shaped pump profiles facing each other [see Figs. 4(b) and
4(d)]. Now, a clear condensate standing wave forms, strongly localized along
the horizontal direction. The mode number of this planar standing wave can
easily be tuned by changing the distance between two lenses or their pump
power [42]. Moreover, the pump polarization of each pump lens can also be
adjusted to design condensate standing waves with intricate polarization
patterns [45]. These results open a pathway in generating structured, high
density, polariton condensates spatially separated from the direct influence
of the reservoir by simply adjusting the geometric configuration and the
excitation power of the nonresonant pump.

##  V Conclusions

In summary, we have theoretically investigated all-optical planar microlensing
techniques on condensates of exciton-polaritons. The lenses are created by
using spatially patterned nonresonant excitation profiles that provide both
gain and blueshift to the polariton modes. We stress that our scheme should
not be confused with that of resonant control [46, 47] where auxiliary
“condensates” are directly injected to provide spatially patterned polariton
blueshift.

We studied the condensate dynamics first in a source-and-lens pump setup as
shown in Fig. 1. We provide a comparison of the rich nonlinear dynamics to
that of linear Schrödinger wave mechanics. Scanning across pump power
parameters, we demonstrate a departure of the condensate steady state,
resembling the linear Schrödinger dynamics, into a stable limit cycle state
characterized by multiple spectral peaks and rapid ($\approx 252$ GHz) density
oscillations in the focal region. This result holds promises for polaritonic
clock generators in integrated circuits [48]. We next studied the condensate
behaviour in a simpler setup consisting of only a single lens shaped pump
driven above threshold. This resulted in highly anisotropic condensate
emission and, amazingly, much stronger focusing of condensate waves outside
the pump region, as compared to the source-and-lens scheme. This last result
opens possibilites in generating polariton condensates that are separated from
detrimental reservoir dephasing effects and might obtain unprecedented
coherence times, favorable for highly sensitive planar matter-wave
interferometers [25].

The possible reservoir devices and their applications are not limited by the
examples we present in this paper, and we hope this work will stimulate the
theoretical and experimental application of reservoir optics in polariton
condensates. Our findings are also relevant to atomtronics [49, 50] where
arbitrary all-optical control over the atom’s potential landscape is possible
[51].

###### Acknowledgements.

The authors acknowledge the support of the UK’s Engineering and Physical
Sciences Research Council (grant EP/M025330/1 on Hybrid Polaritonics), and the
European Union’s Horizon 2020 program, through a FET Open research and
innovation action under the grant agreement No. 899141 (PoLLoC). H.S.
acknowledges the Icelandic Research Fund (Rannis), grant No. 217631-051.
Y.W.’s studentship was financed by the Royal Society
RGF$\backslash$EA$\backslash$180062 grant.

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Reservoir optics with exciton-polariton condensates: Supplemental information

##  S1 Positive meniscus lens in the linear regime

For a double concave shaped potential $V(\mathbf{r})$, we recall the
Lensmaker’s equation in the ray optics limit where the focal length $f$
follows,

$\frac{1}{f}=(n-1)\left[\frac{1}{R_{1}}-\frac{1}{R_{2}}+\frac{(n-1)T}{nR_{1}R_{2}}\right].$
(S1)

Here, $R_{1,2}$ are the radius of curvature of the front and the back surfaces
of the lens, respectively (left and right edges), and $T$ represents the lens’
thickness. In the case of using a positive meniscus lens where $R_{1}>0$, the
thickness $T$ of the lens in Eq. (S1) needs to be taken into account. It is
instructive to define the distance,

$\delta=f\frac{T}{R_{1}}\left(\frac{1}{n}-1\right),$ (S2)

which refers to the distance between the front surface in the positive
meniscus lens and the principal point [see Fig. S1(b)]. The radius of the
front and back surface of the lens is, respectively,
$R_{1}=$14.5\text{\,}\mu\mathrm{m}$$ and $R_{2}=$10.0\text{\,}\mu\mathrm{m}$$,
and the thickness of the lens $T=$4.5\text{\,}\mu\mathrm{m}$$. The radius of
the back surface, the lens’ thickness, and the resonant source is the same one
used in Fig. 2 in main text. For a lens with
$V_{r}=$1.6\text{\,}\mathrm{meV}$$ and $V_{i}=$0\text{\,}\mathrm{meV}$$, we
obtain $\delta\approx$10.0\text{\,}\mu\mathrm{m}$$ and
$f\approx$26.1\text{\,}\mu\mathrm{m}$$ using Eq. (7) in the main text to get
the effective refractive index $n$.

Figure S1 shows the same numerical experiment as Fig. 2 in the main text using
this time a positive meniscus lens. Compared to the results of planoconcave
lens (main text) we observe enhanced scattering of the incident plane wave
front onto to the positive meniscus which results in poorer focused
transmission. This result is in contrast to the case presented in Fig. S3
where a positive meniscus emitter focuses waves more efficiently than a
planoconcave lens.

((a))

((b))

((c))

((d))

((e))

((f))

Figure S1: (a,b) The normalized density, $|\Psi|^{2}$, and (d,e) phase,
$\text{arg}{(\Psi)}$, of the wave function without (a,d) and with (b,e) the
lens potential. The source and the lens are outlined with green solid lines.
(c,f) Line profile of the wavefunction density along the lens axis [$y=0$] as
a function of varying real (c) and imaginary (f) part of the potential. In (b)
to detail the focal region, the value of the normalized density larger than
$0.4$ is saturated in the colormap. In (b) and (e) the yellow dashed lines
indicate the principal plane of the lens and the focal point with distance $f$
and $\delta$ is the distance between the front surface to the principal plane
(the first yellow dashed line from left to right). The green vertical dashed
lines in (c) and (f) indicate the source location and the front and back
surface of the lens and $T$ indicates the distance between two green dashed
lines.

##  S2 Reservoir positive meniscus lenses above threshold

Figures S2(a) and S3 show the same calculation as given in Fig. 4 and 5
(planoconcave lens) in the main text with this time a pump shaped into a
positive meniscus lens. In comparison with the planoconcave lens detailed in
main text, we point out the stronger localization and higher density of the
condensate outside the pump region in the positive meniscus lens.

((a))

((b))

((c))

((d))

Figure S2: (Left column) Condensate density $|\Psi|^{2}$ and (Right column)
phase $\text{arg}{(\Psi)}$ in the steady state. The nonresonant pump is shaped
into a (a) positive meniscus lens showing clear focusing of the emitted planar
waves outside the pumping area, and a (b) positive meniscus resonator made
from two lenses facing each other. Note that each individual lens is below
threshold but the system/resonator as a whole has a lower threshold and thus
supports a standing wave condensate at lower powers. The FWHM of the pump
profiles is outlined in green.

((a))

((b))

Figure S3: (a) Same as Figs. S2(a) for higher lens powers and $\eta=180$ and
$\hbar\xi=84.2g$. (b) Condensate density line profile along $y=0$ for varying
lens power corresponding to panels (a). The vertical green dashed lines
indicate the pump (lens) region.