# Architecture Knowledge Representation and Communication Industry Survey

Haben Birhane Gebreweld  Vrije Universiteit AmsterdamThe Netherlands
<EMAIL_ADDRESS>
(2022)

###### Abstract.

Background:  The literature presents several approaches, such as views,
viewpoint, and architecture decision records (ADRs), to describe software
architectural knowledge (AK). On the other hand, sustainability is a subject
that is receiving increasing attention in software engineering, particularly
in relation to software architecture. However, there appears to be a lack of
industry reviews on these topics from a practical perspective.

Aim:  In this research we aim to understand the current practice in
architecture knowledge, and to explore where sustainability can be applied to
address sustainability in software architecture in the future.

Method: We used a survey, which utilized a questionnaire containing 34
questions and collected responses from 45 architects working at a prominent
bank in the Netherlands, aimed to evaluate the practical representation and
communication of architectural knowledge and sustainability.

Result: Our analysis yielded two primary discoveries and several intriguing
detailed results regarding how AK is captured and conveyed to diverse
stakeholders. The report aims to communicate two essential messages to guide
future research in the field. Firstly, it seems crucial to develop a new
architectural element that connects various architectural features and
perspectives tailored for different stakeholders. Secondly, providing clear
guidance, references, and goals is essential to motivate architects to adopt
Sustainable Software Engineering practices.

Conclusion: After analysing the data collected through this survey, we have
concluded that: a) There are no established domain-specific AK methods/tools
in the financial domain. Most practitioners use domain-generic tools. b) A new
architectural element that links the various architectural features and
viewpoints created for various stakeholders appears to be necessary. c) There
is sufficient sustainability awareness and motivation among software
architects. However, what they lack are clear guidance, references, and goals
to practice sustainable software engineering.

Software Engineering, Architecture Knowledge, Sustainability, Empirical
Experiment,

††copyright: acmcopyright††journalyear: 2022††doi:
10.1145/1122445.1122456††conference: Vrije Universiteit Amsterdam; 14-16 June,
2022; Amsterdam, The Netherlands††booktitle: Vrije Universiteit Amsterdam,
2022, Amsterdam, The Netherlands††price: 15.00††isbn: 978-1-4503-XXXX-X/18/06

##  1\. Introduction

Software architectural knowledge refers to the knowledge acquired while
designing a software architecture, encompassing the assumptions, decisions,
context, and other factors involved in that process (Babar et al., 2009).
Various approaches have been developed both in literature and industry to
depict this knowledge, such as views and viewpoints(Clements et al., 2003),
architecture decision records (ADRs)(Smith, 2020), and standards like ISO
42010(Clements et al., 2003) and the C4 Model(Brown, 2018). However, for this
knowledge to be effective, it is important that all relevant stakeholders
share information about the architecture and how it is represented. The way
this information is communicated depends on the organizational structures
involved and can take various forms, such as wikis, workshops, emails, etc.
Understanding how architectural knowledge is represented and communicated in
professional practice is important to identify appropriate relationships that
address sustainability elements in software architecture. By studying how this
knowledge is represented and shared, we can gain insights into best practices
for ensuring that this knowledge is effectively communicated and can be used
to make informed decisions about the sustainability of software architecture.

As researchers, we develop intriguing methods and techniques for managing
architectural knowledge, while practitioners have their preferred means of
capturing and sharing architectural information. To ensure that our methods do
not become a ”silver bullet” that practitioners do not utilize, it is crucial
to conduct industry reviews. By building upon existing industry practices and
filling in any missing pieces, we can develop effective and useful methods
that practitioners will embrace.

The purpose of this research is to gain insight into the current practices
related to architecture knowledge and explore how sustainability can be
integrated into software architecture in the future. Our objective is to
characterize architecture knowledge and sustainability from the perspective of
software architects, specifically with regards to representation and
communication in the professional practice context. To achieve this, we
conducted a questionnaire survey and gathered responses from architects
working at a prominent bank in the Netherlands about their experiences in the
industry, focusing on how they represent and communicate architectural
knowledge and sustainability. Regarding scientific contributions, as far as we
are aware, our study is the first of its kind to explore how software
architects perceive and utilize sustainability aspects within software
architecture in an industrial context, along with examining architectural
knowledge representation and communication.

This study offers several significant contributions, including:

  * ✓

A practical review of architectural knowledge representation and communication
techniques utilized in the industry.

  * ✓

An assessment of how practitioners approach representing and communicating
sustainability aspects in software architecture.

  * ✓

This study presents a particular collection of AK representation and
communication techniques utilized by software architects who work in the
financial industry.

The paper is structured in the following manner. In section 2, we review
previous studies that have explored the relationship between architectural
knowledge and sustainability in software architecture. Section 3 outlines how
we designed and executed the survey. We present a summary of the survey
results in section 4, and in section 5, we provide a detailed analysis of the
findings. This analysis aims to make sense of the results and convey the main
insights gained from the study. Finally, in section 6, we discus the threats
to validity of our study, and we provide our conclusion in section 7.

##  2\. Related Work

There is a wide range of architectural modeling languages available, but it is
unclear whether they are capable of adequately describing software
architecture to meet users’ requirements. Additionally, the strengths,
limitations, and needs of these languages are uncertain, creating a gap
between what is offered and what users need. Malavolta et al.(Malavolta et
al., 2013) aimed to address this gap by surveying 48 practitioners from 40 IT
businesses in 15 countries in order to plan for the next generation of these
languages. The study examined the perceived benefits, drawbacks, and
requirements related to existing languages. While practitioners were generally
satisfied with the design capabilities of their employed architectural
languages, the study revealed dissatisfaction with the architectural language
analysis features and their capacity to describe extra-functional attributes.
Moreover, the study found that the use of architectural languages in practice
was mostly influenced by industry development rather than academic research,
and there was a need for a more formal and practical architectural language.
Our research shares similarities with the aforementioned study as we, too, are
investigating how architectural knowledge is represented from the perspective
of industry practitioners. To achieve this, we conducted a survey among
software architects working for a leading bank in the Netherlands. Our study
is distinct in that it delves into sustainability and the communication of
architectural knowledge among stakeholders. In addition to exploring a
specific domain (financial domain), we go beyond the use of architecture
description languages and investigate how architects communicate and share
their knowledge.

Despite a significant amount of research and development of various models and
tools, the widespread adoption of Architectural Knowledge Management (AKM) in
software companies is lacking due to the cost of capturing AK(Capilla et al.,
2008) Capilla et al.,. Determining what the industry needs from AK to get
through this barrier and identifying the advantages and disadvantages of the
current AK techniques are therefore necessary. Capilla et al.(Capilla et al.,
2016) undertook an informal retrospective analysis based on their prior work
as researchers and proponents of numerous AK research methodologies in order
to address this. By conducting a series of interviews with various software
businesses, they also looked into the trends and problems for a future
research agenda to support the usage of AK in contemporary software
development methods. They came up with some interesting observations using
this method. While we are also looking into the tools and techniques
practitioners use to capture and communicate architectural knowledge, which
will help us understand current trends in the industry, our study has some
parallels to the research mentioned above. Our research, in contrast to the
aforementioned study, has a keen focus on comprehending how software
architects represent and communicate both architectural knowledge and
sustainability.

While there have been secondary research studies conducted on sustainability
in software engineering, none have particularly addressed software
architecture. Andrikopoulos et al.’s(Andrikopoulos et al., 2022) systematic
mapping study seeks to fill this research gap by exploring the confluence
between sustainability and software architecture. The study’s findings showed
that current studies have neglected the holistic viewpoint required to resolve
such a complex issue by excessively emphasizing on particular sustainability-
related dimensions. To develop the maturity of the field, more reflective
research studies and improved coverage of the architecting life cycle
activities are required. The report suggests a research agenda for
sustainability-aware software architecture based on these findings. Our
research is similar to the study as we also aim to explore the incorporation
of sustainability aspects into software architecture. However, our study takes
a unique approach by focusing on how sustainability aspects of software can be
effectively represented and communicated from the perspective of software
architecture practitioners by conducting industry survey with software
architects.

##  3\. Study design and Execution

By conducting this study, we aim to provide software architects and the
research community with a useful evaluation of how Architecture Knowledge (AK)
and Sustainability are represented and communicated from a practical point of
view. Our research objective is formally characterized by employing the
structure proposed by Basili et al. (Basili et al., 1994) as follows.

Analyze | Architecture Knowledge & Sustainability
---|---
For the purpose of | Characterizing
With respect to | Representation & Communication
From point of view of | Software Architects
In the context of | Professional Practice

Therefore, the present study defines the primary research questions (RQs) and
its corresponding sub-research questions as follows:

  1. $RQ1$

How is software architecture knowledge represented and communicated in
practice?

When addressing RQ1, we investigate and evaluate the entire industrial context
of how AK is represented and communicated. In section 4, we will provide
additional information on the tools, techniques, standards, and documentation
utilized by software architects in the industry. This will aid us in
understanding the industrial structure of AK representation and communication,
which we can exploit to integrate sustainability elements into software
architecture in the future.

    1. $RQ1.1$

How is software architecture knowledge represented in the financial domain?

Our goal through RQ1.1 is to achieve a more comprehensive understanding of the
frequently utilized and advantageous AK elements in the financial domain.
Moreover, we intend to gain insight into the tools utilized in the financial
industry to supplement the AK representation component of RQ1.

    2. $RQ1.2$

How is software architecture knowledge communicated in the financial domain?

Similar to RQ1.1, our objective with RQ1.2 is to gain an understanding of the
tools and techniques utilized by practitioners in the financial industry to
communicate AK effectively.

    3. $RQ1.3$

What architecture knowledge methods are domain-specific or domain-generic?

Our aim with RQ1.3 is twofold: to identify any architecture knowledge methods
that are specific to the financial sector and to identify any domain-generic
methods that can be applied to the financial industry.

  2. $RQ2$

How can sustainability aspects be represented and communicated in software
architecture?

Through RQ2, we aim to gather insights from software architects on how they
would incorporate sustainability aspects into their daily work and software
architecture. Given their wealth of expertise, we aim to challenge software
architects to provide possible ways for integrating sustainability into
software architecture.

Figure 1. Study Design.

We provide a three-phase research methodology in order to respond to these
RQs. Figure 1 shows an overview, which is expanded upon below.

In Step (1), To begin designing the survey, the first step is to identify the
research questions and map them to a set of questions that will be used to
gather information from software architects about their industry practices. To
accomplish this, we have included a variety of questions, including commonly
used ones from the literature that pertain to demographics and architectural
knowledge. This initial step is crucial because it impacts the quality of the
information we obtain from the population. Upon completion, we will have a
comprehensive survey that will be distributed to the significant community of
software architects at one of the largest banks in the Netherlands.

We ask respondents 34 questions in this survey, with 91% of them being open-
ended and allowing respondents to candidly express their unique experiences.
Figure 2 depicts the flow of the eight blocks numbered Q1-Q8. The blocks in
Figure 2 are labeled with the research questions they intend to answer, except
for the consent, demographics, and conclusion blocks. The following blocks are
included:

Figure 2. Survey Questionnaire Flow

Consent: We explained the purpose of the study, which is to understand the
current practice in architecture knowledge, and to explore where
sustainability can be applied to address sustainability in software
architecture in the future. We include the researchers’ contact details as
well as their confidentiality policy.

Demographics (Q1.1-Q1.4): By probing the participants’ professional
backgrounds, particularly their involvement in software projects (see Figure
3), their experiences in their present organizations, and their specific roles
or positions within the organization, this section aims to learn more about
the participants (see Figure 4)

AK in Practice (Q2.1-Q2.5): We start this part with a formal explanation of
our interpretation of architectural knowledge to avoid any misunderstandings
(AK)111Architecture Knowledge (AK) is defined as knowledge elements including
architecture design, design decisions, assumptions, context, and other
elements that together determine why a particular solution is the way it is.
We acknowledge that there may be different ways to understand AK, though. In
this section, we are therefore asking participants about the kinds of AK they
document and retain, the AK components they believe are missing, and the AK
elements they consider to be the most valuable. Our objective in this section
is to comprehend how participants are representing and communicating AK.

AK representation in financial domain (Q3.1-Q3.4): In a similar manner, we
began this section by providing a formal description of our interpretation of
AK representation222Architecture Knowledge (AK) Representation is defined as
capturing and preserving AK in a particular form (e.g., diagrams, PowerPoint,
views, viewpoints, or principles). We ask participants about the notations,
languages, methods, and tools they use to capture AK in general and
specifically in their most recent project, as well as the architectural
documentation they find most useful. As we are specifically targeting
architects who work in a bank, our goal is to gain an understanding of how AK
is represented in the financial domain.

AK communication in Practice (Q4.1-Q4.3): In this section, similar to the one
above, we provide a description of what is meant by AK
communication333Architecture Knowledge (AK) Communication describes to how the
knowledge is disclosed between involved stakeholders (e.g., via workshops, or
corporate sharing platforms). We also inquire with the participants about the
stakeholders involved in their roles, as well as the tools and methods they
use to communicate with different stakeholders. By doing so, we are able to
gather first-hand information on how AK is communicated in practice.

Domain Specific Vs Domain Generic AK (Q5.2-Q5.3): We inquired with the
participants about their familiarity with certain AK methods that are unique
to their specific business domain, as well as the regulations they keep in
mind while representing or communicating AK within their domain. Our goal is
to distinguish between AK tools and methods that are specific to their
business domain and those that have a more general purpose.

Sustainability aspects within software architecture (Q6.1-Q6.5): In this
section of the survey, our goal is to explore how software practitioners
incorporate sustainability aspects into their architectural decisions. To
achieve this, we have included a series of questions designed to better
understand the participant’s perception of IT sustainability. We begin by
asking what the concept of IT sustainability means to the participant.
Following this, we ask architects whether they consider sustainability aspects
during their work, and depending on their response, we delve further to
understand which aspects they consider or the reasons for not considering
them. We also ask whether participants are aware of any IT sustainability
targets set by their company or department, and if they integrate
sustainability aspects into their daily work. Through these questions, we seek
to gain insights into how architects interpret sustainability, both generally
and specifically in the context of software architecture.

Survey Conclusion (Q7.1-Q7.2): Participants are encouraged to provide any
additional information they feel is important in this section. Specifically,
we inquire about what they believe is necessary to accurately represent and
communicate AK, and whether they have any comments about the study itself.
These questions are designed to capture any issues that may be of significance
to participants but have not been addressed in the survey.

In Step (2), we first conduct a pilot survey with a small group of research
and industry experts to check the quality of the survey and eliminate any
possible pitfalls and unclear aspects before reaching out to the main survey
population. This step generates a set of feedback that needs to be
incorporated into the original survey design. Then, we conduct a survey with
the main population consisting of software architects working for a leading
bank in the Netherlands. The objective is to produce a set of architecture and
sustainability representation and communication (ARC) techniques used in
industry. To determine the main population of the survey and ensure our survey
was conducted effectively, we first established the objective of the study,
which was to conduct a practical review of architectural knowledge and how it
is represented and communicated in the financial industry. After analyzing the
possible population for our survey, we determined that software architects
would be the most suitable participants. This is because they possess
extensive knowledge about AK and its application in the industry (check Figure
3), and because the organization already has architects in different software
architecture roles with decades of experience (check Figure 4).

Next, we reached out to the identified population using a two-fold approach.
The first approach involved obtaining a mailing list of software architects
within the organization, while the second involved requesting Team leads of
different structures within the organization to provide us with a list of
architects under their department. By consolidating the information from these
two sources, we generated a draft list of 145 architects. After eliminating
redundant names and removing the architects needed for an extensive interview,
we arrived at a set of 124 architects. Using these two steps to the best of
our abilities, we tried to reach out to all software architects working in the
bank. However, we did not conduct any further steps to verify if we indeed
reached out to all architects.

The survey was conducted using Qualtrics444https://www.qualtrics.com/ and
designed to be anonymous to alleviate concerns about judgment or consequences.
We reached out to the 124 architects via email and received 45 (39%) survey
responses, with eight indicating they no longer worked at the company and 69
not replying.

We compiled all recorded responses from all participants into a spreadsheet
format where each column represents a specific question and each row
represents a response from a participant. However, we made the decision to
make all questions, except for the demographic and consent questions,
optional. As the consent questions are crucial to obtain legal permission from
participants to record their responses, and the demographic data is essential
for interpreting the remaining optional questions. As a result, there are
variations in the responses we received from each participant, as some
attempted to answer all questions, while others chose to skip questions they
did not want to answer.

We analyzed the responses given to each question to find trends and prevailing
viewpoints about the specific topic raised by the question. To facilitate
analysis, we divided the questions into two categories: closed-ended and open-
ended. For closed-ended questions, we simply counted the number of
occurrences. For open-ended questions, we categorized responses by
interpreting each response’s intent(we presented the complete analysis of the
responses in the replication
package555https://docs.google.com/spreadsheets/d/1KIMpIWXGCASJX2WlJROAGwDfn1b9yH_p/edit?usp=share_link&ouid=117224618047071995271&rtpof=true&sd=true).
We created these categories using information from various sources, including
the literature (such as the dimensions of sustainability), the study question,
and similarities in the responses’ intended purposes(check Table 6). Our main
objective was to encode open-ended responses so that quantitative and
conventional statistical analyses (Miles and Huberman, 1994) could be applied
to the data.

The reflection of the results to address our core research questions is the
final step in Step (3). Our objective is to provide a summary of the methods
for architectural representation and communication that are currently used, as
well as how architects address sustainability issues when designing software.
We end by summarizing the takeaways from this practical review.

Figure 3. Survey Population Experience in Years

Based on the data presented in Figure 3, it is evident that 94% of the survey
participants have engaged in software projects for a minimum of 10 years, with
their experience ranging from 10 to 41 years. This suggests that the results
obtained in our study were derived from experts who possess extensive and
valuable experience gained from a long and successful industrial career. Only
two of the respondents reported having less than 10 years of experience, with
7 and 9 years respectively.

We were fortunate to have received participation from a diverse group of
architects in the bank, encompassing a broad range of roles. This enabled us
to gain insights into software architecture from various levels of
abstraction, as well as the experiences of different stakeholders. As
illustrated in Figure 4, our participants spanned 7 different architectural
roles, with the majority of them being Domain Architects (37%) and IT
Architects (26%). The next most frequent roles were Solution Architects (11%)
and Enterprise Architects (8%).

Figure 4.  Current Roles of Participants within the bank.
Where DA stands for domain architect, IA for IT architect, SA for solution
architect, EA for enterprise architect, SWA for software architect, BA for
business architect, and HA for hybrid cloud architect.

##  4\. Results

The main results we inferred from the data666Raw Data of Survey:
https://bit.ly/3EvC2H3 we gathered are presented in this section. It does so
in accordance with the five blocks from Q3 to Q7 and the survey structure
specified in Section 3 of the report.

Table 1 displays the responses to questions in the Q3 block, which focuses on
the practical application of AK. We initiated the block by asking
participants, ”What does AK mean to you?” (Capilla et al., 2016). Most
participants provided definitions that complemented the formal definition we
provided. For instance, one participant explained that ”AK is a group of
elements, such as problem statements, triggers to have a solution, reasons for
creating a new solution, scope, requirements, assumptions, current state
architecture, transition state, and target state. It involves defining and
registering exceptions/deviations to deliver a solution building block.”
However, a few participants shared unique perspectives. One stated that ”AK is
also about knowing what kind of architectural artifacts (e.g., future state
architectures, current state architecture, guidelines, standards) exist in the
organization and identifying any missing artifacts. But most importantly, it
involves interpreting and using them correctly.”

We present the results for the Q4 block, which pertains to the practical
representation of AK, in Table 2. Table 3 displays the results of the Q5
block, which examines the implementation of AK communication. We began by
asking participants about the stakeholders with whom they need to communicate
AK knowledge in their present position. The stakeholders that participants
engage with differ depending on their current role, in addition to their
peers. For example, business architects communicate with business and IT
management, business change teams, and IT delivery teams. Domain architects,
on the other hand, engage with product owners, enterprise architects,
principal architects, developers, business analysts, IT leads, Security CISO,
Design Authority, and external vendors. IT architects communicate with Grid
Owner, Product Owner, Business Analyst, Enterprise Architects, and Domain
Architects.

The results for the Q6 block, which pertain to domain-specific versus domain-
generic AK, are presented in Table 4. Finally, we summarize the results for
the Q7 block, which concerns questions related to sustainability aspects
within software architecture, in Table 5.

Table 1. Architecture Knowledge (AK) representation in financial domain Q2.2 |  Question: What type of AK do you document and keep? Capilla et al.(Capilla et al., 2016)
---|---
Answer: 26 out of 32 respondents, which represents 81% of the participants,
mentioned Solution Intent as the AK to keep and document. Meanwhile, Current
State Architecture was mentioned by 10 participants (31%), and Design
Decisions were mentioned by 25% of respondents. Future State Architecture was
mentioned by 22% of the participants. Five participants mentioned both
Guidelines and Standards as the AK to capture and retain. Additionally, there
were several other AK documents mentioned by the participants that are worth
noting. These include High level Design, Information Architecture,
Architectural Landscape, Problem Statements, and Architectural Review
Comments.
Q2.3 |  Question: Do you capture AK consistently in all projects? Capilla et al.(Capilla et al., 2016)
Answer: Of the 31 respondents to this question, 18 participants (58%) answered
”Yes” while the remaining 13 individuals (42%) answered ”No”. Among the 13 who
answered ”No”, some specified their reasons, including 31% who said that every
project is different, 31% who stated that AK is not required, 15% who believed
that it can sometimes be an overkill, and others mentioned that it is labor-
intensive and that they don’t have enough time.
Q2.4 |  Question: In your experience, what are the AK elements that you miss?
Answer: Of the 31 participants who answered this question, 14% of them
reported being satisfied with the existing AK elements. However, 26% (8
individuals) identified AK elements that act as a means of communication and
bridge between diverse stakeholders as the ones they miss the most in their
work. On the other hand, 23% pointed out that AK elements that give the
context and status of the architecture were the ones they miss in their work..
Additionally, 13% of participants mentioned missing AK elements related to
Clarity, Detail, and Guidance, and 10% mentioned missing elements related to
Design Decisions. Finally, 6% of participants missed AK elements related to
Sustainability Aspects.
Q2.5 |  Question: In your experience, what are the AK elements that you find particularly useful?
Answer: Out of 31 participants who responded to the question, 12 participants
(39%) found AK elements related to Standards, References, and Guidelines to be
the most beneficial. Furthermore, 20% of participants chose Architecture Model
and Business Architecture as the most useful each. Thirteen percent of
participants found Solution Intent777A Solution Intent (SI) describes the high
level, implementation dependent solution architecture and provides guidance to
teams by defining the solution direction. to be the most beneficial, while 10%
each chose Design Decisions and context & status of the architecture as the
most useful AK elements.
Table 2. Architecture Knowledge (AK) representation in financial domain Q3.1 |  Question: Do you know any standard notation or language to capture AK? Capilla et al.(Capilla et al., 2016)
---|---
Answer: Of the 31 participants who responded to the question, 90% (28
participants) indicated ”Yes”. Among these 28 participants, 82% (23
individuals) specified that they use ArchiMate to capture AK, while 5
participants (18%) specified using UML. Additionally, 2 participants each
(totaling 7%) mentioned using Sparx EA, Draw.io, PowerPoint, and BPMN as their
preferred languages and notations for capturing AK.
Q3.2 |  Question:In your experience, what is the most useful architectural documentation? Capilla et al.(Capilla et al., 2016)
Answer: Of the 31 participants who responded to this question, 9 individuals
(29%) identified ArchiMate as the most useful architecture documentation.
Additionally, 5 participants each (totaling 16% of the total) mentioned
Current State Architecture and Diagrams as being useful. Three participants
each (10% of the total) identified Solution Intent, Views/Viewpoints, and
PowerPoint as the most useful. Finally, 2 participants (7% of the total) each
mentioned Design Decision Documents and Visio as being useful.
Q3.3 |  Question: If you think about your last project, in what format was the knowledge stored?Capilla et al. (Capilla et al., 2016)
Answer: 31 participants responded to the question on how they store AK,
mentioning various formats and methods. ArchiMate, Word document, and
PowerPoint were equally popular among 9 participants (representing 29%) as the
formats used to store AK in their last project. Solution Intent was mentioned
by 8 participants (26%), while Confluence was used by 6 participants (19%).
Q3.4 |  Question: What tools or methods do you use to capture and represent AK?
Answer: Of the 31 respondents, 17 individuals (55%) use ArchiMate, 15 (48%)
use PowerPoint, 14 (45%) use Sparx EA, and 11 (35%) use Confluence to capture
and represent AK. Additionally, 10 participants use Microsoft Word and 9 use
Visio for this purpose.
Table 3. Architecture Knowledge(AK) communication in practice  Q4.2 |  Question:If you think of your last project and the AK you have stored, how did you communicate the knowledge to your stakeholders?
---|---
Answer: In their last project, 26 participants responded to the question on
how they communicated AK with different stakeholders. Among the respondents,
38% (10 individuals) used Written Documents to communicate AK, while 35% (9
individuals) mentioned using Meetings and PowerPoint equally. Additionally,
Email was used by 27% (7 individuals) of the participants, and Workshops were
used by 3 individuals.
Q4.3 |  Question: In general, what tools or methods you use to share and communicate AK?
Answer: Out of the 30 individuals who responded to this question, 47% (14
participants) primarily use PowerPoint for communication, 37% (11
participants) use email, 30% (9 participants) use Confluence, and 20% (6
participants) use both meetings and documents.
Table 4. domain-specific Vs generic-specific AK Q5.1 |  Question: Do you know certain methods for AK which are exclusively valid or applied to your business domain?
---|---
Answer: Out of 31 respondents to this question, only 5 (16%) responded “Yes”,
and specified BPMN, TOGAF, SAFe, Target State Design as exclusive AKs to the
finance area.
Q5.2 |  Question: Can you think of any other AK methods that are general-purpose and that have not already been mentioned?
Answer: In response to the question, 20 participants provided feedback. Of
these, 11 participants (representing 55% of the respondents) answered with
”No.” Among the 8 participants who provided an answer to the specific inquiry,
they mentioned the following frameworks: TOGAF, Agile Design Thinking, BPMN,
Service Oriented Architecture, and Standardized Architecture Decision Records
(ARDs).
Q5.3 |  Question: In general, do you have to keep certain regulations in mind (e.g., GDPR, sustainability targets, etc.) while representing or communicating AK in your business domain?
Answer: Out of 30 respondents to this question, 87% (26 participants)
disclosed that they consider certain regulations while representing and
communicating AKs.
Table 5. Sustainability aspects of software architecture  Q6.1 |  Question: What is IT sustainability for you?
---|---
Answer: Among the 23 individuals who responded to the question, 30% (7
participants) demonstrated a comprehensive understanding by referring to at
least two aspects of sustainability or software engineering principles.
Specifically, 39% (9 participants) associated IT sustainability with the
environmental dimension, while 22% (5 participants) focused on the technical
dimensions. The remaining two participants understood IT sustainability in
terms of its economic dimension and its ability to cope with changing
environments across all domains.
Q6.2 |  Question:Are you aware of any IT related sustainability targets or measures in your organization/department?
Answer: Out of 28 respondents to this question, 18 (representing 64% of the
total) reported not being aware of any sustainability targets. The remaining
10 participants (36%) reported having knowledge of some targets, with the most
commonly mentioned targets being cloud-related policies and targets related to
energy consumption, both at 33%.
Q6.3 |  Question: Do you consider sustainability aspects in your current role?
Answer: Of the 28 participants who responded to the question, 75% (21
participants) said ”Yes”. Out of these 21 participants, 48% (10 participants)
considered the technical aspect of sustainability, while 14% (3 participants)
considered the environmental aspect, and 9% (2 participants) considered the
economic aspect. Additionally, 14% (3 participants) considered other aspects
such as business and quality requirements, and adapting to changes.
Q6.4 |  Question: What tools or methods do you use to capture and represent AK?
Answer: Of the 31 respondents, 17 individuals (55%) use ArchiMate, 15 (48%)
use PowerPoint, 14 (45%) use Sparx EA, and 11 (35%) use Confluence to capture
and represent AK. Additionally, 10 participants use Microsoft Word and 9 use
Visio for this purpose. We also inquired about how and where they would
incorporate sustainability into their daily work if it were necessary. Some
suggested including it as a quality attribute in the Solution Intent or
providing guidance on what to assess. Others suggested integrating it into the
business architecture, domain architecture, intake phase, data center, and
patterns and designs.
Table 6. Description of Coding Categories of Qualitative data analysis Q.2.4
In your experience, what are the AK elements that you miss?
---
Category |  Description |  Response Example |  No |  (%)
Communication & Bridge |  AK elements that serve as a bridge between various view/viewpoints intended for different stakeholders and enable effective communication to provide a comprehensive understanding of the architecture. |  ”As architecture is about communication, with different views, we tend to develop views that are understood by architects (which gives grip on the architecture), but less meaningful for (other) stakeholders, linking architecture views to (non architectural) views of different stakeholders is now lacking. We tend to speak our own language and not the language of different stakeholders.” |  8 |  26%
Context & Status |  AK elements that provide actual status of IT components and context of architectural changes |  ”Enterprise level Architectural Fitness Function, that are functions that can be added to the pipelines. So you can validate that what is implemented is in line with the architectural guidelines.” |  7 |  23%
Clarity, Details & Guidelines |  AK elements related to meaning full principles, details, guidelines that apply to the entire IT landscape of the firm. |  ”The detail level of a solution design is always a challenge. Depending on what a team needs, you can decide to include more details. For teams that are mature less details are required. I think it would be good to have guidance on how detailed a SI should be.” |  4 |  13%
Design Decisions |  AK elements related to architectural design decisions. |  ”We have just started focusing on Design Decisions. We should focus more on this. DD captures clear rationale for a specific choice and the other alternatives assessed including pros and cons. This helps a person also understand how a system has evolved and why it has evolved this way. It may point to improvements that are possible today and not when the DD was made.” |  3 |  10%
Sustainability Aspects |  AK elements related to sustainability aspects, targets, and goals. |  ”sustainability aspect of a solution” |  2 |  6%
Q.6.1 What is IT sustainability for you?
Environmental |  Response that focuses on reducing an organization’s impact on the environment such as lowering carbon emission, improving water and air quality. |  ”IT sustainability is taking the effects into account that an IT implementation has on energy consumption and environment. Considering how to reduce this footprint for generations to come. ” |  9 |  39%
Comprehensive |  A description that consists of two or more dimensions of sustainability or principles of software engineering. |  ”Three parts: 1) archtecture in such a way that the solution requires as less resources (and thus energy) as possible and 2)define and develop in such a way that solution can easily be replaced or enhance when requirements change and 3) set-up a sustainable business model.” |  7 |  30%
Technical |  Responses that focused on how software and hardware can be designed and developed to reduce their environmental impact and improve their efficiency |  ”Reduce complexity; reuse of solutions; run systems on-demand; agile architectures” |  5 |  22%

##  5\. Discussion

We conducted a survey with software architects in one of the leading banks in
the Netherlands and identified two main findings, along with a list of
detailed results from analyzing each cluster of questions designed to address
our research question as discussed in section 3. Overall, the survey yielded
valuable insights into the representation and communication of software
architecture knowledge in practice. The findings are:

Finding 1 A new architectural element that links the various architectural
features and viewpoints created for various stakeholders appears to be
necessary.

During the survey, we asked participants about the architectural elements they
felt were missing in their work. This question was found to be revealing as it
elicited many interesting responses. We categorized and analyzed these
responses, which can be seen in Table 6, to facilitate quantitative data
analysis. Most participants identified the need for architectural elements
that can facilitate communication and bridge different viewpoints and features
for various stakeholders. For example, one participant stated, ”As
architecture is about communication, with different views, we tend to develop
views that are understood by architects (which gives grip on the
architecture), but less meaningful for (other) stakeholders. Linking
architecture views to (non-architectural) views of different stakeholders is
now lacking. We tend to speak our own language and not the language of
different stakeholders.” This view expressed by a participant is not isolated,
as shown in Figure 5, where 8 out of 31 respondents (26%) shared similar
views. Another participant expressed the desire for ”more linkage to use non-
architecture viewpoints” to better represent and communicate AK.

Figure 5. Architectural Elements missed by Architects

Finding 2 Clear guidance, references, and goals are necessary to motivate
architects to practice Sustainable Software Engineering.

Our study has yielded significant insights into the sustainability aspects of
software architecture, with a particular focus on representation and
communication. To assess the sustainability awareness of architects, we asked
them to define sustainable IT. Our analysis revealed that one-third of the
participants who responded demonstrated an impressive understanding by
mentioning two or more aspects of sustainability or sustainable software
engineering principles, as depicted in Figure 6. Moreover, other participants
referred to various sustainability dimensions, indicating a high level of
awareness on the subject.

Subsequently, we inquired whether participants were aware of their
organization or department’s sustainability targets, and the majority
responded negatively, highlighting a lack of awareness in this regard.
Nevertheless, when we asked whether they integrate sustainability aspects into
their work, 75% of participants responded affirmatively, which appears to
contradict their lack of awareness of sustainability targets. However, this
discrepancy may be attributed to their advanced level of understanding of the
concept.

Upon further investigation, most participants reported that they were not
incorporating sustainability aspects into their daily work due to a dearth of
clear guidelines, references, criteria, and goals. Additionally, they
expressed a need for more support in bridging the knowledge gap on how to
implement sustainability aspects in their work. Overall, our study underscores
the importance of incorporating clear guidelines, references, criteria, and
goals on sustainability aspects by the organization to leverage the motivation
and high level of sustainability awareness of architects.

Figure 6. Architects understanding of Sustainable IT

Two main research questions and three sub questions served as the framework
for the survey that is the subject of this report. The following provides a
summary of the findings.

RQ1 How is software architecture knowledge represented and communicated in
practice?

During our research, we discovered that architecture knowledge (AK) is
communicated and represented through various documentation and artifacts such
as Solution Intent, Current State Architecture, Design Decisions, and
Guidelines. However, not all projects consistently capture AK, and some
participants mentioned missing AK elements related to communication, context
and status, design decisions, and sustainability aspects. On the other hand,
participants found AK elements related to standards, references, guidelines,
architecture models, and business architecture to be particularly useful.

RQ1.1 How is software architecture knowledge represented in the financial
domain?

Based on the responses, it can be concluded that standard notations or
languages, such as ArchiMate and UML, are commonly used to capture and
represent software architecture knowledge in the financial domain. ArchiMate
was reported as the most commonly used notation. Various tools, including
PowerPoint, Sparx EA, Confluence, Word documents, and Visio, were identified
as useful for capturing and representing architecture knowledge. The most
useful architectural documentation included ArchiMate, current state
architecture and diagrams, solution intent, views/viewpoints, and PowerPoint.

RQ1.2 How is software architecture knowledge communicated in the financial
domain?

The findings suggest that software architecture knowledge in the financial
domain is communicated through various methods and tools, including written
documents, meetings, presentations, email, and workshops. The choice of
communication method may depend on the stakeholder, the level of involvement,
and the complexity of the information. Overall, PowerPoint is the most
commonly used tool for sharing and communicating AK, followed by email and
Confluence. Meetings and documents are also frequently used, with some
participants reporting the use of workshops. However, it is important to note
that the specific methods and tools for communicating software architecture
knowledge may differ depending on the industry or domain.

RQ1.3 What architecture knowledge methods are domain-specific or domain-
generic?

Architecture knowledge (AK) methods are employed in practice through a
combination of domain-specific and domain-agnostic approaches. While most
respondents did not identify any AK elements specific to the finance domain, a
few mentioned techniques such as BPMN, TOGAF, SAFe, and Target State Design as
being exclusive to finance. However, it is worth noting that TOGAF and BPMN
are not solely utilized in finance. Only a few general-purpose AK frameworks
such as Agile Design Thinking, Service Oriented Architecture, and Standardized
Architecture Decision Records (ARDs) were mentioned. This suggests that there
may be a lack of awareness of domain-specific AKs or that many methods are
considered applicable across different domains. Notably, 84% of respondents
did not mention any domain-specific AKs, highlighting the need for further
exploration of the AK methods unique to specific domains.

RQ2 How can sustainability aspects be represented and communicated in software
architecture?

To incorporate sustainability aspects into software architecture, clear
guidelines, references, and goals are required to capture these aspects in
daily work. Participants suggest integrating sustainability into the business
and domain architecture, intake phase, data center, and patterns and designs.
Quality attributes in the Solution Intent can also be used to represent
sustainability aspects. However, a comprehensive understanding of the various
dimensions and principles of sustainable software engineering is necessary for
effective representation and communication. Providing clear guidance,
references, and goals can motivate architects to practice sustainable software
engineering and integrate sustainability into their daily work.

##  6\. Threats to Validity

While we made every effort to conduct a rigorous study, there are several
limitations that must be acknowledged. The external validity of our study may
be limited by the fact that we only targeted participants from a single
organization, despite the fact that we were able to attract a decent number of
participants with significant experience in software architecture. Although
the organization we targeted was large and likely representative of the
industry as a whole, the lack of diversity in our population may limit the
generalizability of our findings. As a result, it may be necessary to
replicate our study with a more diverse sample in order to confirm the
external validity of our results.

The internal validity of a study can be threatened by various factors that
impact the accuracy and reliability of the study’s results. One potential
threat to internal validity in a survey is the use of non-mandatory questions.
In our study, we designed most of the questions to be non-mandatory to avoid
obliging participants to answer questions they may not be qualified to answer
or may have distaste towards. However, this design choice can impact the
overall quality of responses received, as participants may choose not to
answer certain questions, resulting in missing data and potentially biased
results. To address this internal validity threat, we took a careful approach
to analysing the survey responses. Rather than using the total recorded
response for each question, we only considered the total number of respondents
who answered each specific question. By doing this, we were able to account
for missing data and ensure that the responses analysed for each question were
only from participants who chose to answer that particular question. This
approach allowed us to mitigate the potential impact of non-mandatory
questions on the study’s internal validity and ensure that our results were as
accurate and reliable as possible.

Construct validity is a critical aspect of any research study that seeks to
examine and measure theoretical concepts or constructs. In our study, we aimed
to explore the perception of software architecture and architectural knowledge
related to sustainability aspects, and we focused on software architects with
a lot of experience to gather insights. While software architects may be the
ideal candidates to respond to questions related to software architecture, it
can be challenging to determine the best approach for measuring and analyzing
sustainability aspects in software architecture due to the lack of an
established view on the combination of these two areas. As researchers, we
made every effort to define the theoretical concepts and constructs we wished
to study and determine how to measure them in a valid and reliable way.
However, the lack of consensus on the combination of sustainability and
software architecture posed a significant challenge in this regard. Therefore,
we opted to investigate how architects perceive sustainability concepts and
where they may apply sustainability to address sustainability in software
architecture. This approach allowed us to explore the perceptions and
perspectives of experienced software architects, even in the absence of a
well-established theoretical framework for the combination of sustainability
and software architecture. However, this construct validity threat must be
considered when interpreting our study’s findings, and further research is
needed to establish a more robust theoretical foundation for the study of
sustainability in software architecture.

##  7\. Conclusion and Future Work

This paper presents the findings of a survey we conducted on the
representation and communication of architectural knowledge (AK) in practice.
Our study targeted software architects working for a leading bank in the
Netherlands with extensive industry experience in various architectural roles.
We identified two main findings through our analysis of the survey results:
the need for a new architectural element that links different features and
viewpoints created for various stakeholders, and the need for clear guidance,
references, and goals to motivate architects to practice sustainable software
engineering. These findings offer valuable insights for future research in the
field. We recommend further investigation into the development of this new
architectural element and how it can be integrated into existing practices.
Additionally, we suggest exploring ways to promote sustainable software
engineering practices among architects through the establishment of clear
guidance and goals. Our study highlights the importance of effective AK
representation and communication in software industry and the potential
benefits of incorporating sustainable practices into architectural decision-
making.

## References

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  * Basili et al. (1994) Victor R. Basili, Gianluigi Caldiera, and Dieter Rombach. 1994.  The Goal Question Metric Approach.  In _Encyclopedia of Software Engineering_. Wiley, 528–532.
  * Brown (2018) S. Brown. 2018.  The C4 Model for Software Architecture.  https://c4model.com.  Accessed: Jul. 08, 2021.
  * Capilla et al. (2016) Rafael Capilla, Anton Jansen, Antony Tang, Paris Avgeriou, and Muhammad Ali Babar. 2016.  10 years of software architecture knowledge management: Practice and future.  _Journal of Systems and Software_ 116 (2016), 191–205.  https://doi.org/10.1016/j.jss.2015.08.054
  * Capilla et al. (2008) Rafael Capilla, Francisco Nava, and Carlos Carrillo. 2008\.  Effort Estimation in Capturing Architectural Knowledge. In _2008 23rd IEEE/ACM International Conference on Automated Software Engineering_. IEEE, 208–217.
  * Clements et al. (2003) Paul Clements, Felix Bachmann, Len Bass, David Garlan, James Ivers, Reed Little, Paulo Merson, Robert Nord, and Judith Stafford. 2003.  _Documenting Software Architectures: Views and Beyond_.  Addison-Wesley.
  * Malavolta et al. (2013) Ivano Malavolta, Patricia Lago, Henry Muccini, Patrizio Pelliccione, and Antony Tang. 2013\.  What Industry Needs from Architectural Languages: A Survey.  _IEEE Transactions on Software Engineering_ 39, 6 (2013), 869–891.  https://doi.org/10.1109/TSE.2012.74
  * Miles and Huberman (1994) M.B. Miles and M. Huberman. 1994.  _Qualitative Data Analysis: An Expanded Sourcebook_.  Sage Publications, Inc., Thousand Oaks.
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# Exponential localization of moving-end mirror in an optomechanical system

Kashif Ammar<EMAIL_ADDRESS>Department of Electronics, Quaid-
i-Azam University, 45320, Islamabad, Pakistan.  Muhammad Ayub  Department of
Electronics, Quaid-i-Azam University, 45320, Islamabad, Pakistan.  LINAC
Project, PINSTECH, Nilore, 45650, Islamabad, Pakistan.  Farhan Saif
Department of Electronics, Quaid-i-Azam University, 45320, Islamabad,
Pakistan.

###### Abstract

We discuss the dynamics of moving end mirror of an optomechanical system that
consists of a Fabry-Perot cavity loaded with dilute condensate and driven by a
single mode optical field. It is shown that quantum mechanical phenomenon of
dynamical localization occurs both in position and momentum space for moving
end mirror in the system. The parametric dependencies of dynamical
localization are discussed. We also provide a set of parameters which makes
this phenomenon experimentally feasible.

optomechanics, Bose-Einstein condensate, dynamical localization

##  I Introduction

Cavity-optomechanics deals with the interaction of an optical field in a
resonator with confining mirrors Kippenberg . Recent experimental advances
make it possible to couple cold atoms and Bose-Einstein condensates (BEC),
mechanical membrane and nano-sphere with the optomechanical systems Esslinger
. Hence, these hybrid-opto mechanical systems are playground to study
phenomena related to mirror-field interaction, and atom-field interaction
which provide founding principles to develop numerous sensors, and devices in
quantum metrology saifrtm ; saifopt . In opto-mechanics the mechanical effects
of light lead to cooling the motion of a movable mirror to its quantum
mechanical ground state CornnellNat2010 ; TeufelNat2011 ; CannNat2011 , and to
observe strong coupling effects GroeblacherNat2009 ; ToefelNat2011a ;
VerhagenNat2012 . In earlier work optomechanical systems were proposed
precision measurement in gravitational wave detectors Braginsky  and to
measure displacement with large accuracy Rugar , and recently the development
of optomechanical crystals EchenfieldNat2012 . Recent theoretical discussions
and simulations on bistable behavior of BEC-optomechanical system Meystre2010
, high fidelity state transfer YingPRL2012 ; SinghPRL2012 , steady-state
entanglement of BEC and moving end mirror Asjad2011 ; Asjad2012 , macroscopic
tunneling of an optomechanical membrane Buchmann2012  and role reversal
between matter-wave and quantized light field are guiding towards new avenues
in cavity optomechanics. In this submission we report the classical and
quantum dynamics in position and momentum space and show that dispersion is
suppressed in quantum domain while classical Yang ; Zhang  counterpart
increases with time following a power-law. This contrasting behavior is an
analog of Anderson localization and termed as dynamical localization or
exponential localization dynloc . Dynamical localization is studied in
periodically driven nonlinear systems Fishman1982 ; Blumel1991 ; Moore1994 ;
Bardroff1995 ; saifdyn ; Schelle2009  and provides quantum mechanical limits
on classical diffusion of wave packet. The phenomenon of dynamical
localization emerges in the explicitly time dependent system. The single mode
laser field excite condensate atoms to momentum side modes which behave
formally like a moving mirror driven by the radiation force of intra-cavity
field. The side modes induce phase modulation to the field Meystre2013 , which
provide modulation to the radiation pressure exerted on mirror nature1988 . In
this paper, the model of the system is presented in section II. In section
III, we derive the Langevin equation and in section IV, we explain dynamical
localization of the condensates in the system. In section V, we explain
dynamical localization as a function of modulation amplitudes.

##  II The Model

We consider a Fabry-Pérot cavity of length $L$ with a moving end mirror driven
by a single mode optical field of frequency $\omega_{p}$ and BEC with N-two
level atoms trapped in an optical lattice potential Nature2008 ; Science2008 .
Moving end mirror has harmonic oscillations with frequency $\omega_{m}$ and
exhibits Brownian motion in the absence of coupling with radiation pressure.

The Hamiltonian of BEC-optomechanical system is,

$\hat{H}=\hat{H}_{m}+\hat{H}_{a}+\hat{H}_{T},$ (1)

where, $\hat{H}_{m}$ describes the intra-cavity field and its coupling to the
moving end mirror, $\hat{H_{a}}$ accounts for the BEC and its coupling with
intra-cavity field while, $\hat{H}_{T}$ accounts for noises and damping
associated with the system. The Hamiltonian $\hat{H}_{m}$ is given as
LawPRA1995 ,

$\hat{H}_{m}=\hbar\triangle_{c}\hat{c}^{{\dagger}}\hat{c}+\frac{\hbar\omega_{m}}{2}(\hat{p}^{2}+\hat{q}^{2})-\xi\hbar\hat{c}^{{\dagger}}\hat{c}\hat{q}-i\hbar\eta(\hat{c}-\hat{c}^{{\dagger}}),$
(2)

where, first term is free energy of the field,
$\Delta_{c}=\omega_{c}-\omega_{p}$ is detuning, here, $\omega_{c}$ is cavity
frequency and $\hat{c}^{{\dagger}}$ ($\hat{c}$) are creation (annihilation)
operators for intra-cavity field interacting with mirror and condensates and
their commutation relation is $[\hat{c},\hat{c}^{{\dagger}}]=1$. Second term
represents energy of moving end mirror. Here $\hat{q}$ and $\hat{p}$ are
dimensionless position and momentum operators for moving end mirror, such
that, $[\hat{q},\hat{p}]=i$. Intra-cavity field couples BEC and moving end
mirror through radiation pressure. When $n$ photons interacts with the surface
of mirror in cavity round trip time $t=2L/c$, $c$ represents speed of light
and $L$ is cavity length, and transfer $2\hbar k$ momentum to moving end
mirror, here $k=\omega_{c}/c$. Therefore, the radiation pressure force
$\hat{c}^{{\dagger}}\hat{c}\hbar\omega_{c}/L$ will change the position
$\hat{q}$ of moving end mirror and couples moving end mirror with intra-cavity
field. Third term represents this coupling and
$\xi=\sqrt{2}(\omega_{c}/L)x_{0}$ is the coupling strength where,
$x_{0}=\sqrt{\hbar/2m\omega_{m}}$, is zero point motion of mechanical mirror
of mass $m$. Last term gives relation of intra-cavity field and output power
$|\eta|=\sqrt{P\kappa/\hbar\omega_{p}}$, where, $P$ is the input laser power
and $\kappa$ is cavity decay rate associated with outgoing modes.

Now the Hamiltonian for BEC and intra-cavity field and their coupling is
derived by considering quantized motion of atoms along the cavity axis in one
dimensional model. We assume that BEC is dilute enough and many body
interaction effects are ignored Meystre2010 . We have

$\hat{H}_{a}=\frac{\hbar
U_{0}N}{2}\hat{c}^{{\dagger}}\hat{c}+\frac{\hbar\Omega}{2}(\hat{P}^{2}+\hat{Q}^{2})+\xi_{sm}\hbar\hat{c}^{{\dagger}}\hat{c}\hat{Q}$
(3)

here, first term describes energy of field due to the condensate. Second term
expresses the energy of the condensate in the cavity following harmonic
motion. Here, $\hat{P}=\frac{i}{\sqrt{2}}(\hat{b}-\hat{b}^{{\dagger}})$ and
$\hat{Q}=\frac{1}{\sqrt{2}}(\hat{b}+\hat{b}^{{\dagger}})$ are dimensionless
momentum and position operators for condensate which are related as
$[\hat{Q},\hat{P}]=i$ and $\Omega=4\omega_{r}$ where, $\omega_{r}=\hbar
k^{2}/2m_{a}$ is recoil frequency of an atom associated with the change in
energy due to absorption or emission of a single photon. Last term in equation
(3) describes coupling energy of field and condensate with coupling strength
$\xi_{sm}=\frac{\omega_{c}}{L}\sqrt{\hbar/m_{sm}4\omega_{r}}$, where,
$m_{sm}=\hslash\omega_{c}^{2}/(L^{2}NU^{2}_{0}\omega_{r})$ is the effective
mass of side mode which shows that side mode formally behave like a mirror
whose motion is driven by interacting radiation pressure.

##  III Langevin Equations

The Hamiltonian $\hat{H}_{T}$ accounts for the effects of dissipation in the
intra-cavity field, damping of moving end mirror and depletion of BEC in the
system via standard quantum noise operators Noise1991 . The total Hamiltonian
$H$ leads to coupled quantum Langevin equations for position and momentum of
moving end mirror and BEC, viz.,

$\displaystyle\frac{d\hat{c}}{dt}$ $\displaystyle=$
$\displaystyle\dot{\hat{c}}=(i\tilde{\Delta}+i\xi\hat{q}-i\xi_{sm}\hat{Q}-\kappa)\hat{c}+\eta+\sqrt{2\kappa
a_{in}},$ (4) $\displaystyle\frac{d\hat{p}}{dt}$ $\displaystyle=$
$\displaystyle\dot{\hat{p}}=-\omega_{m}\hat{q}+\xi\hat{c}^{{\dagger}}\hat{c}-\gamma^{\prime}_{m}\hat{p}+\hat{f}_{B},$
(5) $\displaystyle\frac{d\hat{q}}{dt}$ $\displaystyle=$
$\displaystyle\dot{\hat{q}}=\omega_{m}\hat{p},$ (6)
$\displaystyle\frac{d\hat{P}}{dt}$ $\displaystyle=$
$\displaystyle\dot{\hat{P}}=-4\omega_{r}\hat{Q}-\xi_{sm}\hat{c}^{{\dagger}}\hat{c}-\gamma_{sm}\hat{P}+\hat{f}_{1m},$
(7) $\displaystyle\frac{d\hat{Q}}{dt}$ $\displaystyle=$
$\displaystyle\dot{\hat{Q}}=4\omega_{r}\hat{P}-\gamma_{sm}\hat{Q}+\hat{f}_{2m}.$
(8)

In above equations $\tilde{\Delta}=\Delta_{c}-NU_{0}/2$, whereas
$\hat{a}_{\mathrm{in}}$ is Markovian input noise of the cavity field. The term
$\gamma^{\prime}_{m}$ gives mechanical energy decay rate of the moving end
mirror and $\hat{f}_{B}$ is Brownian noise operator Pater06 ; Vitali2001 . The
term $\gamma_{\rm sm}$ represents damping of BEC due to harmonic trapping
potential which effects momentum side modes while, $\hat{f}_{1m}$ and
$\hat{f}_{2m}$ are the associated noise operators assumed to be Markovian.

To consider the classical dynamics, we consider positions and momenta as
classical variables. In the steady state, we use adiabatic approximation in
which we set time derivative of the field as zero in equation(4). The steady
state field defined as

$\alpha=\frac{\eta}{\kappa+i(\Delta_{a}-\xi q+\xi_{sm}Q)}.$ (9)

Here, $\hat{c}\rightarrow\alpha$ is the classical limit. We use this relation
in equations (5,7) and put these equations in the second derivative of
equations (6,8). As a result the coupled equations of motion for the moving
end-mirror and BEC reveal their coupled nonlinear dynamics, that is,

$\displaystyle\frac{d^{2}q}{dt^{2}}$ $\displaystyle=$
$\displaystyle-\omega_{m}^{2}q+\frac{\omega_{m}\xi\eta^{2}}{\kappa^{2}+(\tilde{\triangle}+\xi
q-\xi_{sm}Q)^{2}},$ (10) $\displaystyle\frac{d^{2}Q}{dt^{2}}$ $\displaystyle=$
$\displaystyle-4\omega_{r}^{2}Q-\frac{4\omega_{r}\xi_{sm}\eta^{2}}{\kappa^{2}+(\tilde{\triangle}+\xi
q-\xi_{sm}Q)^{2}}.$ (11)

Now we describe Hamiltonian as $H=K+V$ where, $K$ is kinetic energy of the
system and the potential energy $V$ is obtained from equation (10). We further
assume that $\xi_{sm}<<\xi$ which provides the BEC evolution as harmonic
oscillator with frequency $4\omega_{r}$, such that,
$Q=Q_{0}\cos(4\omega_{r}t)$, where, $Q_{0}$ is the maximum displacement of BEC
from mean position. We introduce some dimensionless parameters defined as,
$\gamma=\omega_{m}/\omega_{r}$, $\beta=\eta^{2}/\kappa^{2}$,
$\mu=\Delta/\kappa$, $\mu_{1}=\xi/\kappa$, $\lambda=\frac{\xi_{sm}}{\xi}Q_{0}$
where, $Q_{0}=Q_{1}/\lambda_{p}$ and using dimensionless
time,$\tau=\omega_{r}t$. By using above parameters and integrating equation
(10) we get,

$V=\frac{dq}{d\tau}=\frac{1}{2}\gamma^{2}q^{2}-\int\frac{\gamma\xi\beta/\omega_{r}}{1+[\mu+\mu_{1}\\{q-\lambda\cos(4\tau)\\}]^{2}}dq$
(12)

Now, we can write Hamiltonian $H$,

$H=-\frac{\partial^{2}}{2\partial
q^{2}}+\frac{1}{2}\gamma^{2}q^{2}-\int\frac{\gamma\xi\beta/\omega_{r}}{1+[\mu+\mu_{1}\\{q-\lambda\cos(4\tau)\\}]^{2}}dq$
(13)

By using transformation $x=Q-\lambda\cos(4\tau)$, we find an effective
Hamiltonian for the mirror, that is,

$H_{eff}=\frac{1}{2}\tilde{p}^{2}+\frac{1}{2}x^{2}+x\lambda_{eff}\cos(4\tau)-\gamma_{m}\beta\arctan(\mu-\mu_{1}x),$
(14)

where, $\gamma_{m}=\frac{4\xi}{\gamma\omega_{m}}$ and
$\lambda_{eff}=(1+\frac{32}{\gamma^{2}})\lambda$ is the effective modulation.

In our later work we consider the power of external field $P=0.0164$ mW with
frequency $\omega_{p}=3.8\times 2\pi\times 10^{14}$ Hz and wave length
$\lambda_{p}=780$ nm. Coupling of external field and intra-cavity field is
$\eta=18.4\times 2\pi$ MHz and frequency of intra-cavity field is considered
$\omega_{c}=15.3\times 2\pi\times 10^{14}$ Hz which produces recoil of
$\omega_{r}=3.8\times 2\pi$ kHz in atoms placed in cavity of length
$l=1.25\times 10^{-4}$ m and having decay rate $\kappa=1.3\times 2\pi$ MHz.
Moving end mirror of cavity should be perfect reflector that oscillates with a
frequency $\omega_{m}=\omega_{r}/k^{-}=15.2\times 2\pi$ kHz. The mirror-field
and condensate-field coupling strengths are, respectively, $\xi=15.07$ MHz and
$\xi_{sm}=14.39$ kHz. Detuning of the system is taken as
$\Delta=\Delta_{c}+\frac{U_{0}N}{2}=0.52\times 2\pi$ MHz, where vacuum Rabi
frequency of the system is $U_{0}=3.1\times 2\pi$ MHz and number of ultra cold
atoms placed in the BEC-optomechanical system are $N=2.8\times 10^{4}$
Esslinger ; Carmon05 ; Carmon07 ; AIP ; Ferdinand . These experimental
parameters are used to obtain numerical results presented in all the figures.

##  IV Dynamical localization of Mirror

In order to discuss classical dynamics, we solve Hamilton’s equations and in
quantum domain, we solve time dependent Schrödinger equation. The classical
dynamics of mirror is studied by Poincaré surfaces of section for different
modulation amplitudes.

Figure 1: Phase Space (Poincaré surface of section) plots of moving end mirror
for different effective modulation amplitudes $\lambda_{eff}=2.6\times
10^{-6}m,5.2\times 10^{-6}m,1.05\times 10^{-5}m,3.15\times 10^{-5}m$ in figure
(a), (b), (c) and (d), respectively. Where, $\mu=-0.4$, $\mu_{1}=2$,
$\beta=1.8$ and $\gamma_{m}=0.6034$.

Fig-1 shows Poincaré surfaces of section for $500$ initial position of the
mirror. For sufficiently large modulation classical phase space shows mixed
behavior i.e. stable islands are emerged in chaotic regions. We note that
chaotic regions increase with the modulation amplitude while resonant regions
shrink. As dynamical localization emerges from quantum evolution of wave
packet in classically chaotic regime, we study the behavior of momentum and
position dispersion of movable mirror for modulation amplitude showing mixed
phase space behavior. The quantum dynamics of the mirror is explored by
evolving a Gaussian wave packet, $\Psi(x)$, defined at $\tau=0$,

$\Psi(x)=\frac{1}{\sqrt{\sqrt{2\pi}\Delta
x}}\exp(-\frac{(x-x_{0})^{2}}{2\Delta
x^{2}})\times\exp(-i\frac{p_{0}x}{k^{-}}),$ (15)

where, initial dispersion in position and momentum is defined in such a way
that it satisfy the minimum uncertainty condition. For all numerical
calculations, we use $\Delta x=1$, $\Delta p=0.5$ and scaled Planck’s constant
is $k^{-}=1$. Whenever classical parameters are evaluated for comparison, we
use a Gaussian ensemble with the same initial dispersion in position and
momentum and initially placed with same mean position and momentum as in the
case of quantum wave packet.

Figure 2: Classical and quantum dispersion in position, $\Delta x$ (upper
figure) and momentum $\Delta p$, (lower figure) of movable vs time at
$\lambda_{eff}=1.05\times 10^{-5}m$. The remaining parameters are the same as
in Fig-1.

Fig 2 shows classical and quantum dispersion in position and momentum space as
a function of time. The dispersion in momentum space follow classical
diffusion for quantum break time $t\sim 12.7~{}msec$ and after this time
$t>12.7~{}msec$, the quantum mechanical dispersion of mirror in momentum space
becomes saturated and oscillate around a mean value. While classical
dispersion increases continuously with time, following $t^{\alpha}$ law, where
$\alpha=0.6$ showing anomalous diffusion in classical domain. Similarly,
quantum dispersion in position space initially diffuses for quantum break time
$t\sim 23.7~{}msec$ and later saturates and show small oscillations around
mean value whereas, classical dispersion in position space follows a continues
diffusion even after the quantum break time. The suppression in quantum
dispersion in position and momentum space is a signature of dynamical
localization of mirror in position and momentum space SchliechPRL1997 .

Figure 3: Classical and quantum mechanical momentum distribution at time
$t=131.5~{}msec$ in momentum space $W(p)$ (upper figure) and position space
$W(x)$ (lower figure). The insets show exponential localization behaviour in
distribution on logarithm scale. The remaining parameters are the same as in
Fig-1.

We study position and momentum distribution of the wave packet for
$t=131.5~{}msec$ which is a time at which dispersion in position and momentum
is dynamically localized. Fig-3 shows classical and quantum mechanical time
averaged probability distribution of mirror both in position,
$W(x)=|\Psi(x)|^{2}$ and momentum space $W(p)=|\Psi(p)|^{2}$. The inset show
the quantum results in log scale. These results show that classical
probability distribution in position and momentum space is broader than the
quantum mechanical distribution. Classical probability distribution in
position and momentum space is almost equally distributed over the entire
space. While, quantum mechanical probability distribution both in position and
momentum space is maximum near $x=1$. The peaks appearing in quantum
mechanically position and momentum distribution are due to stable islands in
the chaotic sea as shown in Fig-1. The quantum mechanical distribution of wave
packet in position and momentum space at large time $t=131.5~{}msec$ is
confirmation of dynamical localization of the mirror. The spatiotemporal
behavior of mirror for $\lambda_{eff}=4$ is shown in Fig-4 for the same
parameters as in Fig-2. The time evolution of probability distribution of
quantum wave packet shows localization effects both in position and momentum
beyond quantum break time. The quantum probability distribution in position
and momentum space remains localized with small fluctuations in position and
momentum space while, in classical domain distributions show ever spreading
behavior in position and momentum space.

Figure 4: Spatio-temporal dynamics of mirror in momentum and position space vs
time shows: (a) quantum mechanical probability distribution in momentum space,
(b) classical distribution in momentum space, (c) quantum mechanical
probability distribution in position space, and (d) classical distribution in
position space. The remaining parameters are the same as in Fig-1.

##  V Effects of modulation

In this section, we study the effects of modulation on the position and
momentum dispersion of movable mirror. Fig-5 shows effects of modulation on
classical and quantum dispersion of the mirror in position and momentum space.
The classical ensemble or quantum wave packet is evolved for $t=131.5~{}msec$
for fixed rescaled Planck’s constant $k^{-}=0.1,~{}1$ and dispersion at
$t=131.5~{}msec$ is plotted versus fixed effective modulation. We observe that
when there is no modulation, classical and momentum dispersions show not a
significant difference both in position and momentum space. But as effective
modulation increases, the quantum dispersion shows dynamical localization
effects while classical dispersion in position and momentum shows diffusion.
The localization effects appear for the effective modulation where
corresponding classical phase space is mixed. For small effective modulation,
corresponding phase space is near integrable and no dynamical localization
effects appear while for large $\lambda_{eff}$, localization effects are seen.
We get strong dynamical localization at that particular effective modulations
where the difference between classical dispersion and quantum mechanical
dispersion is maximum and and vice versa is also true.

Figure 5: Dispersion in momentum and position space vs modulation at
$t=131.5~{}msec$. Upper figure shows the behavior of momentum dispersion vs
effective modulation amplitude $\lambda_{eff}$ while lower figure shows the
behavior of position dispersion vs $\lambda_{eff}$. Quantum dispersion
behavior is plotted for rescaled Planck’s constant $k^{-}=0.1,0.5,1$. The
remaining parameters are the same as in Fig. 1

The Fig-5 also shows behavior of quantum dispersion in position and momentum
space for different values of effective Planck’s constant, $k^{-}$. As $k^{-}$
is decreased curve representing quantum dispersion approaches to the curve
showing classical dispersion which is a manifestation of correspondence
principle.

##  VI Conclusion

We conclude that the motion of moving end mirror weakly coupled with Bose-
Einstein condensate in the optomechanical cavity (Fabry-Perot Cavity) shows
dynamical localization in position and momentum space. Moreover, for
increasing modulation dynamical localization in momentum space doesn’t
disappear as in the case of driven optical lattice Bardroff1995  where this
localization effect is governed by Bessel function with modulation strength as
argument of Bessel function and dynamical localization revives for the value
of arguments where Bessel function is zero. In position space, localization
length slowly increases with modulation as shown in the Fig-5. The set of
parameters used here makes this phenomenon experimentally feasible.

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