Vidby is a Switzerland-based multi-product IT company specializing in technologies of understanding and development of AI-powered translation solutions. The company has coined the term "Technologies of Understanding" as part of its unique approach. Vidby is in the constant process of developing an array of innovative solutions that help people communicate on a global scale. Vidby is a Google Cloud Technical Partner & YouTube’s Recommended Vendor for content localization. PRODUCTS & SOLUTIONS 🎥 Vidby Video Translation – the company’s flagship product - AI-powered platform for fast and accurate video translation and dubbing. Recommended by YouTube. Free plan is available. 👥 Vidby MeetUp – an AI application for live translation of offline/online meetings & live broadcasts. Download from App Store / Google Play 🧑🏻💻 Vidby Call Translator – a real-time translation solution for Google Meet. Zoom and Microsoft Teams are upcoming. Multiple participants. 140+ languages and dialects. 🕵🏼 YouTube Audio Tracks Detector by vidby is an extension that sends users notifications when new videos and/or audio tracks in chosen languages are added to their favorite YouTube channels. 💰 Vidby Video Investment Platform – a platform where investors and content creators come together to craft captivating projects and bring ambitious ideas to life. 🙌🏻 Vidby Workspace – vidby’s marketplace and working platform for freelance language professionals (transcribers, translators, pre-editors, post-editors) and dubbing actors. 📝 Vidby Document Translation – AI-powered solution for online translation of documents in 100+ languages. MISSION: To make any content available to everyone while preserving the world’s linguistic diversity.

vidby

Recently, Hollywood directors and actors have been protesting and blaming artificial intelligence (AI) for the loss of jobs in their industry. They fear that the advancement of AI technology will render them unemployed. However, history has proved that such fears are often unfounded. And this time, AI can be a valuable tool rather than a threat, ultimately leading to a brighter future for everyone involved. Let's take a closer look at several historical cases when technological advancements have actually created more jobs and improved productivity. Automation and Accounting At the beginning of the 2000s, there was significant concern about the automation of accounting and the potential job losses for accountants. However, over the years, the number of accountants has increased tenfold. Automation has become a means of improving productivity and accuracy rather than replacing human accountants entirely. Technology has become a human partner, not a replacement. The Impact of the Assembly Line When Henry Ford introduced the assembly line, many feared massive job losses and reduced human participation. The increased efficiency and productivity resulting from the assembly line led to higher production volumes and increased demand for workers. Instead of soaring unemployment rates, the expanding industry faced a shortage of labor to meet its demands. The assembly line drove economic growth and opened up new avenues for employment. The Emergence of Television Back in 1929, critics claimed that television would eliminate books and theater, proclaiming that there would be nothing left but only television. Almost a century later, we can see that this fear was baseless. The rise of television didn't cause theaters or books to vanish. Instead, it created a new medium of entertainment and generated hype around TV shows. New technologies often complement existing ones, leading to the expansion of opportunities rather than their extinction. The Internet and YouTube As the Internet and platforms like YouTube began to gain traction, concerns about technology taking over resurfaced. Many predicted the death of traditional television. Still, after 17 years, television continues to thrive alongside online video platforms. In fact, television networks have learned to embrace platforms like YouTube, creating synergistic relationships that benefit both industries. Thanks to technological advancements, we're now experiencing a more diverse and dynamic entertainment landscape. The Impact of Word Processors Once upon a time, stenographers were in high demand, with entire departments dedicated to typing up texts on manual typewriters. With the advent of word processors and computers, fears arose that these professionals could disappear. While the profession did disappear, people adapted and transitioned into more valuable and productive roles. The technology allowed them to handle tasks beyond mere transcription, ultimately enhancing their capabilities and creating new professions. AI Creating More Job Opportunities AI integration into the entertainment industry still raises concerns among creators. But they needn't worry. AI has the potential to enhance the creative process, revolutionize how films and TV shows are dubbed and translated, and create opportunities for new employment. Currently, video translation and dubbing require significant labor and time, often leading many creators to give up on the localization of their video content. The solution lies in AI-powered tools like , which are capable of translating and dubbing videos into 70 languages with high quality within 24 hours. vidby Vidby and similar tools have the potential to increase translation volumes hundreds of times. Yet, human involvement will remain crucial because, with no human input, achieving top-notch quality won't be possible in the foreseeable future. With the implementation of AI-powered tools, AI-human translation will thrive and create more job opportunities in the localization market. On top of that, creators can leverage AI to generate more content, reach larger audiences, and ultimately achieve greater demand for their creativity. As a result, more jobs in AI programming, data analysis, and content curation will appear. Creators shouldn't view AI as a threat but embrace it as a valuable tool to boost human creativity and productivity. The key to a brighter future for all lies in humans adapting and evolving alongside technology.

Why AI Is Not the Enemy: Embracing Technology for a Brighter Future

The accelerating rate of adoption of AI in daily life has become a beloved topic in tech publications since the release of ChatGPT, but the trend really began in 2020 with the beginning of the pandemic, when businesses needed to work smarter to survive and started widely adopting AI analytics. Now AI has entered mainstream consciousness, and according to a from the Pew Research Center, people in the US feel slightly more concern than excitement. recent report In its report, “How Americans Feel About AI,” Pew shows that only 18% of Americans are “more excited” about the incorporation of AI in daily life, while nearly double (37%) are more concerned than excited, and 45% equal parts concerned and excited. In Web2 and Web3, it’s important to realize that less than 1/5 of people in the dominant center for tech innovation on Earth are willing to embrace the technology without trepidation. Making AI work for us Audiences develop relationships with apps – not underlying technology. AI is already all around us, helping us get work done more efficiently. The underlying fear is largely concern over losing control of the use of the technology and it’s vast potential to track, catalogue, and commodify our information – exactly the kind of abuse we have come to expect as users from adware to Facebook and Google. But rarely do we expect AI solutions to breakdown barriers between people. That’s what AI-driven translation services are doing. To use a tech metaphor, AI-driven translation services solve one of the interoperability issues between humans who are using different languages which makes it hard for them to interact. We can find simple AI-driven text translation on Google – but according to a recent report by Sandvine, 65% of online traffic. Vidby is an AI and human-driven translation service that allows people to share critical and even across language-barriers, not just with subtitles but by accurately in the language of the target audience. I spoke with co-founder CEO, and Chairman of Vidby, Eugen von Rubinberg about the challenges and potential of video translation to connect all people on Earth. life-saving information dubbing videos Justin: Is the language barrier one of the great challenges to our otherwise connected world? “Language barrier is certainly one of the biggest problems in our world. But there will soon be no more language barriers. Everyone will understand everyone in the world. EVR: With our technologies we wish to create understanding for people all over the world. We see this as part of our mission.” Justin: Why did Vidby decide to focus specifically on video? “Today, video is the best and most common and effective way of conveying information. The volume of consumption and effectiveness of video content is growing every year. It's a big trend that affects all areas. But that's not our only product. In the coming months we will be launching a professional translator for on-site meetings.” EVR: Justin: Has Vidby changed your own perspective on language? “We have been studying this topic intensively for a long time and we can say with certainty that language strongly influences the way people think. It is therefore important, among other things, to preserve linguistic diversity in the world, because this diversity moves the world. That is why we champion linguistic diversity.” EVR: Justin: Will we ever have a figurative 'babelfish' app that translates languages in real-time? “Yes, we already have such a product on iTunes called "vidby Meetap."It’s a handy application to translate personal conversations in real time. Within the next few months we will launch the second version, for iPhone and Android, for simultaneous translation of live broadcasts and live conferences in 70 languages.” EVR: Justin: How are improvements in AI helping you solve new problems? “AI and machine learning is of course at the heart of Vidby, but we also research and develop voices for children, teenagers, and seniors in all languages, for example. We currently don't know anyone else who offers that. EVR: We also solved the issue of correct transfer of gender when translating from one language to another. For example, the phrase "I did" sounds the same for males and females in English, but it sounds different in German. We have also put a lot of effort into developing a technology that can manage the same pronunciation in different languages. These are just a few examples of what we are doing and have done today. This year we plan to develop many more technologies that will close virtually all cases for good voice acting and translation. The only thing we're still working on, but not quite there yet, are emotions in the voice acting. This problem is relevant for the industry in the next few years, but it will also be solved, because technologically it is completely clear how to do it, it just takes time and money.”

AI Video Translation is Connecting the World & Preserving Language Diversity

Shingling in the Generative AI Era

Ukraine's Rapid Push to Deploy AI-Enabled Drones for Battlefield Supremacy

Deep Lake, a Lakehouse for Deep Learning: Conclusions, Acknowledgement, and References

গভীর লেক, গভীর শিক্ষার জন্য একটি লেকহাউস: উপসংহার, স্বীকৃতি, এবং উল্লেখ

Deep Lake, ein Lakehouse für Deep Learning: Schlussfolgerungen, Danksagungen und Referenzen

Deep Lake, una casa de lago para el aprendizaje profundo: conclusiones, reconocimientos y referencias

Deep Lake, un Lakehouse pour le Deep Learning : conclusions, remerciements et références

डीप लेक, डीप लर्निंग के लिए एक लेकहाउस: निष्कर्ष, स्वीकृति और संदर्भ

ディープラーニングのためのレイクハウス、Deep Lake: 結論、謝辞、参考文献

딥 러닝을 위한 레이크하우스인 Deep Lake: 결론, 감사의 말씀, 참고 자료

Deep Lake, um Lakehouse para Deep Learning: Conclusões, Reconhecimento e Referências

Deep Lake, домик у озера для глубокого обучения: выводы, признательность и ссылки

Deep Lake, Derin Öğrenme için Bir Göl Evi: Sonuçlar, Teşekkür ve Referanslar

Deep Lake, Lakehouse cho Deep Learning: Kết luận, Lời cảm ơn và Tài liệu tham khảo

Deep Lake，深度学习的 Lakehouse：结论、致谢和参考文献

Closing the Skills Gap: Addressing the Surge in Demand for Digital Marketing Experts

Gartner predicts generative AI will become a general-purpose technology akin to the steam engine, electricity, and the internet. While the initial hype will fade, its impact will increase as innovative applications are discovered in daily work and life. Look at some of their predictions for AI, particularly Generative AI’s adoption: As entire industries, enterprises, and digital transformation specialists race to adopt this new tech, there are critical factors that must be taken into account. Improper adoption of technology can lead to disastrous results for the business and poor satisfaction for customers and employees who have to use them. I want to share what those factors are in this article. Why Integrate GenAI?
How to Implement GenAI?
What to Use GenAI For? Why Integrate GenAI? Value Proposition The primary reason to consider integrating GenAI is the substantial value it can bring to your business. GenAI has the potential to revolutionize various aspects of your operations, ultimately driving growth and efficiency. Top-Line Benefits One of the most significant advantages of GenAI is its ability to increase user engagement with your target segment, which can lead to higher revenue or new revenue streams altogether. By leveraging GenAI, you, as a business, can create more personalized and interactive content and enhance your customer experience and satisfaction with your products and services. For example, GenAI can help summarize and synthesize large amounts of information quickly, making it easier for users to find what they need and stay engaged with your platform. Bottom-Line Savings On the other hand, GenAI can also lead to significant cost reductions. Automating repetitive tasks such as customer support queries and data entry, allows your team to focus on more strategic activities. AI-powered tools can support your help center, providing instant responses to common questions and reducing the need for extensive human intervention. How to Implement GenAI? Building the GenAI Tech Stack Successful integration of GenAI requires a systematic approach. This involves Applications and use cases that align with your business goals


Choosing the right AI models to power these applications


Using the right infrastructure to deploy these applications and models


Handling the Implementation responsibly without harming trust and security while also upskilling employees and educating customers on how to use GenAI. Implementation and Application To start, you need to identify the areas within your business that will benefit most from AI integration. This could include content generation for marketing and sales, content conversion for streamlined communication, and productivity tools for various teams. Content Generation: Tools like Jasper, Writer.com, Byword, Adobe Firefly, and Midjourney can enhance your marketing efforts by generating high-quality content.


Content Conversion:  Platforms like Wondercraft (text to podcast), Vidyo (long-form video to short clips), Descript (podcasting), and ElevenLabs (AI voices) can transform how your marketing teams operate.


Spreadsheets and Slides: For financial and R&D teams, AI tools like Numerous.ai, Equals, Gamma, and Tome can automate data management and presentation tasks.


General Productivity: Tools like Vowel for meeting transcription and Metaview for interview recording can optimize daily operations. Ensuring Trust and Security Integrating AI also necessitates maintaining high standards of trust and security. Implement robust reporting mechanisms to allow users to flag AI-generated content that may be inaccurate or controversial. This helps build user trust and maintain content quality. Moreover, ethical training and fine-tuning of AI models are crucial. Ensure that user data is collected and used transparently and with explicit consent. Handle all user interactions responsibly to protect privacy and build confidence in your AI systems. What to Use GenAI For? Enhancing Employee Productivity Deploying AI applications can significantly boost employee productivity across various departments. As previously mentioned above, in the implementation and application layer of the GenAI tech stack, these are some use cases for employee productivity. Marketing and Sales: Tools like Jasper and Adobe Firefly can assist in generating compelling content for campaigns, while platforms like Byword enhance SEO efforts.
Content Conversion: AI tools such as Wondercraft and Vidyo can transform text and video content, making it more versatile and accessible.
Spreadsheets and Slides: AI-powered tools like Numerous.ai and Gamma streamline data processing and presentation creation, aiding teams in finance and R&D.
General Productivity: Vowel provides meeting transcription, while Metaview offers optimized recording and summarization for interviews. Poised serves as a personal communication coach, enhancing overall communication skills.
Customer Support: AI tools like Chatbase can automate customer service, providing immediate responses to queries and improving customer satisfaction.
Specific Applications for HR and Hiring Managers: For HR teams, GenAI tools like Metaview and Poised can revolutionize the recruitment process by recording and summarizing interviews, as well as coaching candidates on communication skills.
Customer Support: In customer support, tools like Chatbase can handle routine queries efficiently, freeing up human agents to tackle more complex issues. Integrating GenAI into your business can unlock new levels of efficiency, engagement, and cost savings. So consider the value it brings, the implementation process, and the specific applications for your business, and you can ensure a successful integration of GenAI that drives your business forward. Embrace the future of AI to stay competitive and innovative in today’s rapidly evolving landscape. Hope that helps! Gartner predicts generative AI will become a general-purpose technology akin to the steam engine, electricity, and the internet. While the initial hype will fade, its impact will increase as innovative applications are discovered in daily work and life. Look at some of their predictions for AI, particularly Generative AI’s adoption: As entire industries, enterprises, and digital transformation specialists race to adopt this new tech, there are critical factors that must be taken into account. Improper adoption of technology can lead to disastrous results for the business and poor satisfaction for customers and employees who have to use them. I want to share what those factors are in this article. Why Integrate GenAI? How to Implement GenAI? What to Use GenAI For? Why Integrate GenAI? How to Implement GenAI? What to Use GenAI For? Why Integrate GenAI? Value Proposition Value Proposition Value Proposition The primary reason to consider integrating GenAI is the substantial value it can bring to your business. GenAI has the potential to revolutionize various aspects of your operations, ultimately driving growth and efficiency. Top-Line Benefits Top-Line Benefits Top-Line Benefits One of the most significant advantages of GenAI is its ability to increase user engagement with your target segment, which can lead to higher revenue or new revenue streams altogether. By leveraging GenAI, you, as a business, can create more personalized and interactive content and enhance your customer experience and satisfaction with your products and services. For example, GenAI can help summarize and synthesize large amounts of information quickly, making it easier for users to find what they need and stay engaged with your platform. Bottom-Line Savings Bottom-Line Savings Bottom-Line Savings On the other hand, GenAI can also lead to significant cost reductions. Automating repetitive tasks such as customer support queries and data entry, allows your team to focus on more strategic activities. AI-powered tools can support your help center, providing instant responses to common questions and reducing the need for extensive human intervention. How to Implement GenAI? Building the GenAI Tech Stack Building the GenAI Tech Stack Building the GenAI Tech Stack Successful integration of GenAI requires a systematic approach. This involves Applications and use cases that align with your business goals Choosing the right AI models to power these applications Using the right infrastructure to deploy these applications and models Handling the Implementation responsibly without harming trust and security while also upskilling employees and educating customers on how to use GenAI. Applications and use cases that align with your business goals Applications and use cases that align with your business goals Choosing the right AI models to power these applications Choosing the right AI models to power these applications Using the right infrastructure to deploy these applications and models Using the right infrastructure to deploy these applications and models Handling the Implementation responsibly without harming trust and security while also upskilling employees and educating customers on how to use GenAI. Handling the Implementation responsibly without harming trust and security while also upskilling employees and educating customers on how to use GenAI. Implementation and Application Implementation and Application Implementation and Application To start, you need to identify the areas within your business that will benefit most from AI integration. This could include content generation for marketing and sales, content conversion for streamlined communication, and productivity tools for various teams. Content Generation: Tools like Jasper, Writer.com, Byword, Adobe Firefly, and Midjourney can enhance your marketing efforts by generating high-quality content. Content Conversion:  Platforms like Wondercraft (text to podcast), Vidyo (long-form video to short clips), Descript (podcasting), and ElevenLabs (AI voices) can transform how your marketing teams operate. Spreadsheets and Slides: For financial and R&D teams, AI tools like Numerous.ai, Equals, Gamma, and Tome can automate data management and presentation tasks. General Productivity: Tools like Vowel for meeting transcription and Metaview for interview recording can optimize daily operations. Content Generation: Tools like Jasper, Writer.com, Byword, Adobe Firefly, and Midjourney can enhance your marketing efforts by generating high-quality content. Content Generation: Tools like Jasper, Writer.com, Byword, Adobe Firefly, and Midjourney can enhance your marketing efforts by generating high-quality content. Content Generation: Content Conversion:  Platforms like Wondercraft (text to podcast), Vidyo (long-form video to short clips), Descript (podcasting), and ElevenLabs (AI voices) can transform how your marketing teams operate. Content Conversion :  Platforms like Wondercraft (text to podcast), Vidyo (long-form video to short clips), Descript (podcasting), and ElevenLabs (AI voices) can transform how your marketing teams operate. Content Conversion Spreadsheets and Slides: For financial and R&D teams, AI tools like Numerous.ai, Equals, Gamma, and Tome can automate data management and presentation tasks. Spreadsheets and Slides : For financial and R&D teams, AI tools like Numerous.ai, Equals, Gamma, and Tome can automate data management and presentation tasks. Spreadsheets and Slides General Productivity: Tools like Vowel for meeting transcription and Metaview for interview recording can optimize daily operations. General Productivity : Tools like Vowel for meeting transcription and Metaview for interview recording can optimize daily operations. General Productivity Ensuring Trust and Security Ensuring Trust and Security Ensuring Trust and Security Integrating AI also necessitates maintaining high standards of trust and security. Implement robust reporting mechanisms to allow users to flag AI-generated content that may be inaccurate or controversial. This helps build user trust and maintain content quality. Moreover, ethical training and fine-tuning of AI models are crucial. Ensure that user data is collected and used transparently and with explicit consent. Handle all user interactions responsibly to protect privacy and build confidence in your AI systems. What to Use GenAI For? Enhancing Employee Productivity Enhancing Employee Productivity Enhancing Employee Productivity Deploying AI applications can significantly boost employee productivity across various departments. As previously mentioned above, in the implementation and application layer of the GenAI tech stack, these are some use cases for employee productivity. Marketing and Sales: Tools like Jasper and Adobe Firefly can assist in generating compelling content for campaigns, while platforms like Byword enhance SEO efforts. Content Conversion: AI tools such as Wondercraft and Vidyo can transform text and video content, making it more versatile and accessible. Spreadsheets and Slides: AI-powered tools like Numerous.ai and Gamma streamline data processing and presentation creation, aiding teams in finance and R&D. General Productivity: Vowel provides meeting transcription, while Metaview offers optimized recording and summarization for interviews. Poised serves as a personal communication coach, enhancing overall communication skills. Customer Support: AI tools like Chatbase can automate customer service, providing immediate responses to queries and improving customer satisfaction. Specific Applications for HR and Hiring Managers: For HR teams, GenAI tools like Metaview and Poised can revolutionize the recruitment process by recording and summarizing interviews, as well as coaching candidates on communication skills. Customer Support: In customer support, tools like Chatbase can handle routine queries efficiently, freeing up human agents to tackle more complex issues. Marketing and Sales: Tools like Jasper and Adobe Firefly can assist in generating compelling content for campaigns, while platforms like Byword enhance SEO efforts. Marketing and Sales: Content Conversion: AI tools such as Wondercraft and Vidyo can transform text and video content, making it more versatile and accessible. Content Conversion: Spreadsheets and Slides: AI-powered tools like Numerous.ai and Gamma streamline data processing and presentation creation, aiding teams in finance and R&D. Spreadsheets and Slides: General Productivity: Vowel provides meeting transcription, while Metaview offers optimized recording and summarization for interviews. Poised serves as a personal communication coach, enhancing overall communication skills. General Productivity: Customer Support: AI tools like Chatbase can automate customer service, providing immediate responses to queries and improving customer satisfaction. Customer Support: Specific Applications for HR and Hiring Managers: For HR teams, GenAI tools like Metaview and Poised can revolutionize the recruitment process by recording and summarizing interviews, as well as coaching candidates on communication skills. Specific Applications for HR and Hiring Managers: Customer Support: In customer support, tools like Chatbase can handle routine queries efficiently, freeing up human agents to tackle more complex issues. Customer Support: Integrating GenAI into your business can unlock new levels of efficiency, engagement, and cost savings. So consider the value it brings, the implementation process, and the specific applications for your business, and you can ensure a successful integration of GenAI that drives your business forward. Embrace the future of AI to stay competitive and innovative in today’s rapidly evolving landscape. Hope that helps!

3 Things to Consider Before Adding GenAI to Your Business

Design and Deploy a Completely Private and Secured Networking Architecture on Cloud Using Automation

How Useful are Computational Models to Traders in Their Decision Making?

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Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: References

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References Appendix D: QMPE in region (e): 3rd order EP The region (e) in the (d,˜ Γ) plane of Hatano’s model is composed of a single point It is a third order EP where all the non-zero eigenvalues become equal and their eigenvectors coalesce. Since this is a single point with fixed values of ˜d and Γ, the right eigenvectors are explicitly given by Of course |r1i and hℓ1| have same expressions both in Eqs. (3) and (D2). The left eigenvectors are ) The non-zero eigenvalue of the Lindbladian [Eq. (2)] in region (e) is −iλ = −i4√ 3. Importantly, as a consequence of the 3rd order EP, we can only achieve an approximately diagonal form i.e. Jordan normal form LbJn leading to the equation Due to the off-diagonal nonzero defects in Eq. (D5), the time evolution of density matrix elements and other observables pick up additional algebraic terms in time. 1. Density matrix elements In region (e), the density matrix elements have the expressions below In Fig. 15(a), we present existence of t˜+ and t˜− as function of the control parameters ˜dI and ˜dII using the fixed dissipation protocol. The solutions’ surfaces in Fig. 15(a) reveal that there are regions where only one of the solutions t˜+ or t˜− exist leading to single QMPE, and in other regions both the solutions exist resulting in double QMPE. Particularly, lower values of ˜dI and ˜dII correspond to single intersection time solution and therefore single QMPE. To understand the behaviors of t˜+,− more clearly, we fix ˜dI and display t˜+,− as functions of ˜dII in Fig. 15(b). The Fig. 15(b) shows that the two branches t˜+ and t˜− interchange as ˜dI < ˜dII. More specifically, for ˜dI > ˜dII we have t˜+ > t˜− and for ˜dI < ˜dII we have t˜+ < t˜−. The underlying reason is that the discriminant p γ 2 1 − 4γ2γ0 in Eq. (D8) is proportional to sign of ( ˜dI − ˜dII). Thus the two intersection times interchange their roles by changing the sign accompanying the discriminant as we go from ˜dI > ˜dII to ˜dI < ˜dII. The Fig. 16(a) demonstrates an explicit example of double QMPE in ground state probability in region (e). Like other regions, here also the second intersection (shown in the inset of Fig. 16(a)) is much feeble in comparison to the first intersection. 2. Average energy For the same parameters used to obtain QMPE for ρgg(t) in Fig. 16(a), we would like to examine the possibility of QMPE in average energy E(t). Indeed, we observe in Fig. 16(b) that energy also exhibits double QMPE, marked by the twice intersection of zero by the energy difference ∆E(t) between the two copies. The insets of Fig. 16(b) shows separately the two intersections in the time evolution of the energies for the two copies. The second intersection is much weaker in magnitude compared to the first intersection just like the case of ground state probability. Note that the first intersection between the energies of I and II occur after they overshoot their steady state value which they reach during some early time of the relaxation process. 3. von Neumann entropy Here we investigate the von Neumann entropy SvN(t), starting from two copies I and II. We use same parameters that have produced double QMPE both in ground state probability [Fig. 16(a)] and average energy [Fig. 16(b)]. In fact, the Fig. 17(a) shows that the entropy also exhibits double QMPE indicated by twice zero intersections (the second one shown in inset of Fig. 17(a)) of the entropy difference ∆SvN(t) between the two copies. 4. Temperature In this section we study the thermal QMPE in region (e). We observe thermal QMPE in region (e) by using variable dissipation protocol in Fig. 17(b). There we observe single intersection and reversal of identities (i.e. hotter becoming colder and vice versa) occur after the negative temperatures have happened. So, such temperatures can be considered as well-defined and therefore this intersection in Fig. 17(b) corresponds to single QMPE in temperature in region (e).

Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: Appendix D: QMPE in Region (e)

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References Appendix C: QMPE in region (b): purely exponential relaxation In this Appendix we discuss QMPE in region (b) of Hatano’s model that contain eigenvalues 0, −iλ2, −iλ3, −iλ4 obeying 0 < λ2 < λ3. 1. Density matrix elements where In Fig. 12(a), we present the intersection time solutions denoting QMPE in ρgg(t) as functions of the control parameters ˜dI and ˜dII. Indeed, we observe that there are parameter regions where both t˜+ and t˜− are positive and finite, indicating double QMPE. Other regions have only one intersection time, either t˜+ and t˜−, implying single QMPE. To demonstrate double QMPE, the Fig. 12(b) exhibits a specific example of double intersection in ρgg(t). However, the two copies become close enough so that the second intersection (shown in inset of Fig. 12(b)) is weak in magnitude and hard to detect. 2. Energy We look for solutions to ∆Es(t) = 0, which can give to single or multiple QMPE depending on the number of positive finite intersection time solutions. The Fig. 13(a) shows that the average energy exhibits double QMPE indicated by the twice zero intersections of ∆Es(t), for the same set of parameter values producing double QMPE in ρgg(t) in Fig. 12(b). 3. von-Neumann entropy Here we ask if the entropy, like density matrix and energy, also exhibit double QMPE for the same set of parameters used in Figs. 12(b) and 13(a). However, in Fig. 13(b), we observe that the entropy difference ∆SvN(t) between two copies intersect zero only once, giving rise to single QMPE instead of double QMPE. 4. Temperature In Fig. 14(a), we present the time evolution of temperatures for the two copies I and II, one initially starting as hotter with respect to the other. We observe that the temperatures of the two copies intersect each other i.e. the hotter becomes colder and vice versa. Notably, unlike ground state probability [Fig. 12(b)], energy [Fig. 13(a)] that can exhibit double QMPE, we have observed only single QMPE in temperature. If observation of multiple QMPE is possible or not with purely exponential relaxations, remains an open question. 5. Kullback-Leibler divergence Here we study the possibility of QMPE in KL divergence for the same parameters used to explore thermal QMPE in Figs. 14(a). The Fig. 14(b) shows QMPE in KL divergence, however the intersection time corresponds to negative temperatures in Fig. 14(a).

Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: Appendix C: QMPE in Region (b)

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References Appendix B: Eigenvectors of L in region (a1) (a1) In this Appendix, we give the explicit expressions for the right eigenvectors |rki and left eigenvectors hℓk| of the Lindbladian [Eq. (2)] in region (a1) [Sec. V], where k = 2, 3, 4. The right and left eigenvectors corresponding to the zero eigenvalue i.e. steady state. have already been provided in Eq. (3). Below are the expressions for the right eigenvectors The left eigenvectors are given by where

Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: Appendix B: Eigenvectors

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References VI. SUMMARY We have studied QMPE in a two-level quantum system subjected to an oscillatory electric field and dissipative coupling with the environment. The Lindbladian of this two-level driven dissipative system, termed as Hatano’s model, has rich variety of eigen-spectrum, especially EPs and complex eigenvalues which are generic features of non-Hermitian quantum systems. We have investigated the roles of complex eigenvalues and EPs on the QMPE. The exact analytical formulae obtained for time evolved density matrix elements enable us to propose analytical criteria for the QMPE in observables. Interestingly, at the EPs, the density matrix elements and consequently observables (e.g. energy) attain additional algebraic time dependent relaxations apart from the usual exponential relaxations. Such additional algebraic time dependence at the EPs leads to double QMPE (twice intersections between trajectories starting at different initial conditions and relaxing towards the same steady state). We have procured the exact expressions for intersection times of trajectories, that mark the occurrence of QMPE and we provide analytical criteria that demarcate control parameters’ regimes with single or double or no QMPE. Whereas the multiple QMPE at EPs in the present model is restricted to maximum double QMPE i.e. two intersections, the multiple QMPE at the complex eigenvalues can result in even more number of intersections in a oscillatory manner. Notably, we have established the existence of thermal QMPE (single QMPE at EPs and multiple QMPE at complex eigenvalues), in presence of coherence in Hatano’s model. It is fascinating that, similar to energy and entropy, temperature also exhibits multiple QMPE where the initially hotter and colder copies intersect each other multiple times. Such multiple reversals of identities (i.e. hotter becoming colder and vice versa) result in multiple thermal QMPE. In spite of achieving multiple QMPE in several observables (ground state probability, energy, entropy, temperature) using either additional algebraic time dependence at EPs or oscillations at complex eigenvalues, surprisingly the KL divergence (a distance measure from steady state) shows single QMPE only. Overall, our theoretical analysis exhibits that EPs and complex eigenvalues can be used as systematic routes to generate multiple QMPE. However, we should mention that the presence of EPs or oscillations are not necessary for multiple QMPE. In this connection, we have shown that multiple QMPE can be achieved even with purely exponential relaxations, by putting suitable constraints on the gap statistics of the eigen-spectrum (Appendix C). An important and generic future goal in QMPE research would be to understand its utility in speeding up quantum processes. In fact, a recent work on accelerating relaxation in open quantum systems by engineering EPs [84] hints towards the connection of such acceleration to QMPE. Our present work analyzes in detail how EPs lead to QMPE and provide criteria on control parameters to observe the QMPE. It is important to find out the deeper connections between the QMPE and faster relaxations at EPs and for generic open quantum systems. In this connection, an intriguing question is to find how the quantum speed limit may affect the degree of faster relaxation that can be achieved through the QMPE. Our work on QMPE by controlling the amplitudes of oscillatory electric fields hints towards possible experimental realizations of the QMPE in qubit, keeping in mind the recent advancement in qubit related experiment [119]. The analysis of QMPE in the present study paves path for future investigations of QMPE in quantum many-body systems. - We thank Naomichi Hatano, Sosuke Ito, Ryo Hanai, Arnab Pal, Naruo Ohga and Kohei Yoshimura for useful discussions. This work is partially supported by the Grants-in-Aid for Scientific Research (Grant No. 21H01006 and No. 20K14428). A.K.C. gratefully acknowledges postdoctoral fellowship from the YITP. The numerical calculations have been done on Yukawa-21 at the YITP. Acknowledgements.

Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: Summary

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References V. QMPE IN REGION (a1): OSCILLATIONS In this section, we explore QMPE in region (a1) of Hatano’s model which contains complex eigenvalues with both non-zero real and imaginary parts. The eigenvalues are non-degenerate and they are denoted by 0, λ2, λ3 = λre+iλim, λ4 = λre−iλim, where the subscripts re and im correspond to real and imaginary parts, respectively. In addition, the eigenvalues satisfy the relation 0 < λ2 < λre meaning λ2 is the slowest relaxation mode. In contrast to the nontrivial approximate diagonalization [Eqs. (10)- (11)] for second-order EP in Sec. IV, here the complete diagonalization can be obtained through the usual method A. Density matrix elements The the individual density matrix elements ρj (t) (j = 1, 2, 3, 4) are given by To discuss QMPE in the density matrix, let us focus on the ground state probability i.e. ρgg(t). Of course, there is no algebraic time dependence in the relaxation observed in Sec. IV. Rather, the interesting part of the relaxation in region (a1) originates from the complex nature of the eigenvalues and the corresponding coefficients ak in Eq. (27). Specifically, we have found that a3 and a4 are complex conjugates of each other, they take the forms a3 = are + iaim and a4 = are − iaim. Using such expressions, we obtain the following formula for ρgg(t): In Fig. 7 we present an example of multiple QMPE in the ground state density matrix element ρgg(t), where we have chosen parameters that satisfy the QMPE criteria Eq. (30) for region (a1). We observe that the two copies I and II relax in oscillatory manner and intersect each other multiple times. The inset explicitly shows that the difference between the two copies ∆ρgg(t) intersects zero multiple times, confirming the multiple QMPE. Comparing Fig. 3(a) [region (d)] with the inset of Fig. 7 [region (a1)], we understand that the number of intersections in region (a1) is not restricted to two as in region (d). Therefore, in region (a1) one can have many more intersections and higher order QMPE compared to the maximum double QMPE in the region (d). Since the amplitudes of the oscillations can be controlled by tuning the initial driving strengths ˜dI and ˜dII, complex eigenvalues seem to be useful to obtain multiple intersections and thereby multiple QMPE. It would be important in the future to find the exact number of solutions to ∆ρgg(t) = 0 either analytically or numerically. B. Energy The formal expression for average energy E(t) is given by Eq. (18). In the previous Sec. IV, we have already shown that QMPE in ground state probability does not necessarily imply QMPE in energy, in general. However, it is possible to find control parameters that can give rise to QMPE in both observables, although the intersection time(s) are generally different. Following the same path, we would like to see if E(t) shows multiple QMPE for the same set of parameters used to achieve multiple QMPE for ρgg(t) in Fig. 7. Indeed, Fig. 8 exhibits multiple intersections between time evolved copies of energies starting from I and II, both relaxing towards the same final energy. The inset of Fig. 8 alternatively demonstrates the multiple QMPE through multiple intersections of the energy difference ∆E(t) = EI(t)−EII(t) with zero, remarkably different from Fig. 3(b) with a maximum of two such intersections in region (d). C. von Neumann entropy Next we consider the von Neumann entropy SvN(t) [Eq. (22)]. To observe QMPE in SvN(t), we consider the intersection times for the entropy difference between two copies ∆SvN(t) = S I vN − S II vN. In Fig. 9, we present the behavior of ∆SvN(t) for the same set of parameter values used in Figs. 7 and 8. Indeed, we observe that ∆SvN(t) intersects zero multiple times showing the existence of multiple QMPE in the von Neumann entropy. D. Temperature Here, we study the thermal QMPE in region (a1) using the system temperature defined in Eq. (23). In Fig. 10, we present the time evolution of the temperatures of two copies I and II, one initially hotter than the other. It is fascinating that the temperatures of both the copies relax in oscillatory manner towards the same steady state temperature and during the relaxations, they intersect each other multiple times, leading to multiple thermal QMPE. This, in turn, implies that the hotter and colder copies reverse their identities multiple times (after each intersection the previous hotter copy turns into colder copy and remains so until the next intersection). Such oscillatory multiple thermal QMPE is observed here for the first time in quantum systems (as far as our knowledge is concerned), although multiple thermal MPE in classical inertial suspensions has already been reported [42]. The physical interpretation and utility of multiple thermal QMPE have to be explored in future in more details. Note that we have used the same parameter set for multiple thermal QMPE as in Figs. 7-9. Similar to Ref. [65], here also we observe negative temperature [114– 118] in the early stage of relaxation. This is due to nonmonotonic natures of energy and entropy as functions of time. However, it is satisfying that even if we neglect such initial negative temperatures, we observe multiple QMPE at later times of the relaxation process where the temperatures of both the copies remain positive. E. Kullback-Leibler divergence In this subsection we explore the QMPE in KL divergence defined in Eq. (24). Since we have been able to obtain multiple QMPE in region (a1) in all other observables considered here, it is natural to ask about the possibility of multiple QMPE in DKL(t) as well. However, we have found that the KL divergence does not show any intersection between two copies I and II for the same parameter values used in Figs. 7-10. Therefore, we have to use different values of the initial drives (still within the fixed dissipation protocol) to achieve QMPE in DKL(t), presented in Fig. 11(a). Intriguingly, we have only observed single intersection between the two time evolved copies, resulting in single QMPE in KL divergence. This means that once the initially higher distant copy intersects the initially lower distant copy producing QMPE, thereafter the new lower distant copy (that was initially higher distant) remains closer to the steady state for the rest of the relaxation process. In fact, for all other parameter sets that we have checked, we could not find multiple QMPE in DKL(t). The proof of the possibility or impossibility of multiple QMPE in KL divergence in presence of complex eigenvalues remains unsolved and could be an important question to investigate later. In this scenario, it would be interesting to examine the corresponding behaviors of the speeds vKL(t) [Eq. (25)] of the two copies I and II. In Fig. 11(b), we present the time evolution of speeds for the same set of parameters that gave rise to single QMPE in DKL(t). It is fascinating to observe that the speeds, unlike the distances, intersect each other multiple times producing multiple QMPE. The first intersection between the speeds of the two copies [Fig. 11(b)] shows a time lag in comparison to the intersection between distances of the same copies [Fig. 11(a)]. Such a time lag between the QMPE in KL divergence and the speed is in agreement with the observation in region (d) [Fig. 6]. However, contrary to single QMPE in vKL(t) in region (d), we have obtained multiple QMPE in the speed in region (a1). The contrasting behavior of single QMPE in DKL(t) and multiple QMPE in vKL(t) imply that the initially higher distant copy that becomes lower distant after the intersection in distances, can remain at lower distance even if its speed sometimes become smaller than the speed of the current higher distant copy. This is interesting because the system almost behaves like an intelligent and efficient runner in a race who knows how to adjust the speed (i.e. sometimes running faster and sometimes running slower than the people behind) and still remains in the first position to finish the race.

Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: QMPE in Region (a1)

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References Appendix A: Eigenvectors of L in region (d) In this Appendix, we give the explicit expressions for the right eigenvectors |rki and left eigenvectors hℓk| of the Lindbladian [Eq. (2)] in region (d) [Sec. IV], where k = 2, 3, 4. The right and left eigenvectors corresponding to the zero eigenvalue i.e. the steady state. have already been provided in Eq. (3). Below are the expressions for the right eigenvectors where the superscript T denotes transpose. The left eigenvectors are obtained to be where the elements lkn are explicitly given by where the eigenvalues λ2 and λ4 are given by Eq. (12) and the parameters η, δ1, δ2, δ3 used in Eq. (A3) have the following forms

Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: Appendix A: Eigenvectors

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References III. DIFFERENT REGIMES IN d˜− Γ˜ PLANE AND MULTIPLE QMPE Indeed, in the space of control parameters ( ˜dI, Γ˜ I, ˜dII, Γ˜ II; ˜d, Γ), we find ˜ multiple QMPE where the two trajectories corresponding to some observable, intersect each other multiple times before reaching the same steady state. Specifically, we have obtained two or more intersections in a oscillatory manner, in regions (a1) and (a2) with complex eigenvalues. Whereas, at most two intersections or double QMPE can be found in regions (c), (d) and (e) with EP. Interestingly, even number of intersections (e.g. double QMPE or quadruple QMPE) imply that the initial identities of the two copies (higher valued copy and lower valued copy of an observable, more explicitly hotter and colder copies if the observable is temperature) are restored finally after the last intersection. Therefore, in such cases the initially hotter or higher valued copy remains hotter or higher valued after all the intersections have occurred. On the other hand, odd number of intersections (single QMPE or triple QMPE) indicates the ultimate reversal of the initial identities after the final intersection. In these cases the initially hotter or higher valued copy finally becomes colder or lower valued copy after the completion of all intersections. This classification has important consequences when one is not only interested in faster relaxations or intersections at intermediate times, but also interested in faster reach to final or steady state. It would be intriguing in future to find connections between such odd or even QMPE and faster approach to steady state. Notably, if we restrict on monotonic time volution For that case, even number of QMPE is of no use. Only odd number of QMPE can help to reach faster to steady state using QMPE. However, more conditions are required like avoiding overshoot etc. to use odd QMPE to get faster reach to the steady state. In the next sections, we discuss multiple QMPE in different regimes in detail. In this paper, we illustrate QMPEs in region (d) for quench onto the second-order EP in Sec. IV (with more analytical details in Appendix A) and region (a1) for the quench onto the point with complex eigenvalues in Sec. V (with more analytical details in Appendix B). We do not discuss QMPE in regions (c) and (a2) separately, because the analyses of QMPEs in these regions are similar to those in regions (d) and (a1), respectively. The QMPEs in other regions are discussed in the appendices. Appendix C discusses QMPE in region (b) with purely exponential relaxations and Appendix D contains the description of the QMPE at the third-order EP in region (e).

Multiple Quantum Mpemba Effect: Exceptional Points and Oscillations: Different Regimes

This paper is available on arxiv under CC 4.0 license. Authors: (1) Amit Kumar Chatterjee, Yukawa Institute for Theoretical Physics, Kyoto University & Department of Physics, Ramakrishna Mission Vidyamandira; (2) Satoshi Takada, Department of Mechanical Systems Engineering and Institute of Engineering; (3) Hisao Hayakawa, Yukawa Institute for Theoretical Physics, Kyoto University. Table of Links Introduction Model and Protocol Different Regimes in d˜− Γ˜ Plain and Multiple QMPE QMPE in Region (d): Second-order Ep QMPE in Region (a1): Oscillations Summary Appendix A: Eigenvectors of L in region (d) Appendix B: Eigenvectors of L in region (a1) Appendix C: QMPE in region (b): purely exponential relaxation Appendix D: QMPE in region (e): 3rd order EP References IV. QMPE IN REGION (d): SECOND-ORDER EP A. Density matrix elements Using Eqs. (10) and (11) in Eq. (1), we have obtained the following expressions of the density matrix elements ρj (t) (j = 1, 2, 3, 4): where the initial density matrix elements ρn(0) before quench are given by Eq. (5) and the elements ℓk,n are provided in Appendix A. Interestingly, in the presence of the second-order EP in region (d), the density matrix elements in Eq. (13) have picked up an additional algebraic term proportional to time t. This is remarkably different from the time evolution in regions (b) which has only purely exponential relaxations [see Appendix B] and region (a1) that contains damped oscillations [see Sec. V]. The linear t dependence originates from the off-diagonal defect in the Jordan normal form in Eq. (11). To investigate QMPE, we focus on the ground state density matrix element ρgg(t) (≡ ρ4(t)). To detect the QMPE, we would like to find the intersections of the two copies I and II during their relaxation, i.e. solution(s) to the equation ∆ρgg(t) = 0. We obtain the following form of ∆ρgg(t) where the co-efficients α1, α2, α3 are given by The solutions to such transcendental equation in Eq. (14) are given by the Lambert W function, where the equation yey = x has only one solution y = W0(x) if x > 0 and two solutions y = W0(x) and y = W−1(x) if −1/e 6 x < 0. Indeed, such properties of Lambert W function imply that there can be more than one solutions to ∆ρgg(t) = 0, meaning multiple QMPE. More precisely, at most two intersection times i.e. double QMPE can be observed in region (d). Below we provide the criteria for double QMPE and single QMPE in this region: The asymptotic expansions of the functions W0(x) and W−1(x) can be found in Ref. [105]. Note that Lambert W functions have found applications in several fields such as viscous flows [106], crystal growth [107], epitaxial film growth [108], phase separation of polymer mixtures [109], criticality in temporal networks [110] etc. In Fig. 1, we plot the solutions t˜0 and t˜−1 [Eq. (17)] as functions of the driving strengths before quench. Indeed, we observe that there are regions with single QMPE where only t˜0 exists and there are other regions where both solutions prevail leading to double QMPE. In particular, we observe double QMPE for large values of ˜dI and ˜dII. Note that QMPE appears to be unavoidable in the ground state probability ρgg(t) in Fig. 1 i.e. there is no such region in the (d˜ I, d˜ II) where there is no QMPE. B. Energy Here we analyze QMPE in average energy E(t) = Tr[ρb(t)H] where H is the system Hamiltonian presented as a 2 × 2 matrix with elements H11 = 1, H22 = 0, H12 = H21 = ˜d/2 [95]. Consequently, E(t) can be expressed as Note that the average energy directly captures the effect of coherence due to the presence of the off-diagonal density matrix elements in Eq. (18). To explore the QMPE in E(t), we consider the time evolution of the energies starting from the two different initial conditions I and II and look for their intersections during the relaxations. More precisely, we would like to find the solutions to ∆E(t) = 0 where ∆E(t) = EI(t) − EII(t). We obtain the following expression where the elements α1, α2, α3 are given by Eq. (15). Note that Eq. (20) is a similar kind of transcendental equation as Eq. (14), only with different coefficients. Therefore, we can analyze the possibility of QMPE in energy using Lambert W functions as we did for the ground state probability. The two possible solutions to ∆E(t) = 0 are given by where the superscript E denotes energy and is used to differentiate the solutions from Eq. (17) obtained for ground state probability. Figure 2 exhibits the solutions in Eq. (21) as functions of the control parameters ( ˜dI, ˜dII). We observe region with single QMPE (only t˜E 0 exists) as well as region for double QMPE (both t˜E 0 and t˜E −1 exist). In addition, we observe regions where both the solutions are absent i.e. no QMPE. This crucial difference in Fig. 2 compared to Fig. 1 indicates that QMPE in ground state probability does not necessarily implies QMPE in average energy. However, Figs. 1 and 2 also show that there are common parameter values that lead to QMPE both in ρgg(t) and E(t). We present an explicit example in Fig. 3 where both of these quantities exhibit double QMPE, for the same set of parameters. However, the two intersection times for ρgg(t) and E(t) are different from each other. The intersection times for ρgg(t) are comparatively far from each other with the second QMPE being much weaker in magnitude [Fig. 3(a)], whereas the intersection times for E(t) are comparatively much closer to each other, with both the first and second QMPE being comparable in magnitude [Fig. 3(b)]. C. von Neumann entropy Next we consider the von Neumann entropy SvN(t), which is a function of ρb(t), defined as The nonlinearity of SvN(t) as a function of the density matrix elements poses difficulties to perform analytical treatment of the QMPE in von Neumann entropy, in contrast to what we could achieve previously for ground state probability and energy. To observe QMPE in SvN(t), we consider the intersection time(s) for the quantity ∆SvN(t) = S I vN −S II vN. In Fig. 4, we present the behavior of ∆SvN(t) for the same set of parameter values used in Fig. 3. Indeed, we observe that ∆SvN(t) intersects zero twice, implying the existence of double QMPE in the von Neumann entropy, similar to ρgg(t) and E(t). D. Temperature We use the definition of temperature T (t) following Ref. [65] to examine the possibility of thermal QMPE in Hatano’s model, as given below where E(t) and SvN(t) are defined in Eqs. (18) and (22), respectively. The QMPE in system temperature has been observed recently in Ref. [65] using a semi-classical approach. Naturally, it would be interesting to explore thermal QMPE in Hatano’s model that involves the effect of off-diagonal density matrix elements. However, we find that there is no thermal QMPE for the same set of parameters that have produced QMPE for ground state probability [Fig. 3(a)], energy [Fig. 3(b)] and entropy [Fig. 4]. To search for thermal QMPE, we consider a variable dissipation protocol that allows us to use Γ˜ I 6= Γ˜ II 6= Γ˜ (wider parameter space compared to the fixed dissipation protocol Γ˜ I = Γ˜ II = Γ). Indeed, using such protocol, we ˜ have obtained the thermal QMPE shown in Fig. 5 where we present the temperature difference ∆T (t) between two copies I and II. We observe that ∆T (t) becomes zero at some intermediate relaxation time that denotes the intersection of two copies, after which the initially hotter copy becomes colder, leading to thermal QMPE. E. Kullback-Leibler divergence The Kullback-Leibler (KL) divergence is a measure of difference from steady state which is a monotonically decreasing function of time, and it is given by To obtain QMPE in KL divergence, we use the variable dissipation protocol. In Fig. 6(a) we present an example where the copy starting at a higher distance from steady state relaxes faster compared to the other copy with lower initial distance. The copies I and II intersect each other leading to QMPE in DKL(t) and the copy starting at higher initial distance reaches the steady state faster. We have not observed multiple QMPE in KL divergence. It would be an interesting future question to understand if the absence of multiple QMPE in DKL(t) is its generic feature and if the reason is its monotonicity. An important variable of interest in any relaxation process, be it classical or quantum, is the speed of the relaxation. Since the KL divergence gives the distance measure from steady state, a relevant relaxation speed can be defined as where the negative sign is used to complement the fact that the KL divergence decreases with time. It would be interesting to see if the QMPE in DKL(t) is also manifested in the speed vKL(t). For the same set of parameters that produced QMPE in KL divergence [Fig. 6(a)], we investigate QMPE in the speed in Fig. 6(b). Indeed, we observe that the speeds of the two copies I and II intersect each other, leading to QMPE. The higher distant copy goes towards steady state with a higher speed and after the intersection its speed becomes lower. Notably, the intersection times for DKL(t) and vKL(t) are different. In fact, there is a time window where the initially higher distant copy becomes lower distant copy but still maintains its higher speed for some time. It would be intriguing to investigate whether there is some general upper bound on the relaxation speed vKL(t). We end this section by mentioning that the analysis of the QMPE in region (c) would follow similar manner as in region (d) discussed elaborately in the present section. This is because region (c) is also a line of second-order EP just like region (d), the only difference is that the arrangement of eigenvalues for region (c) obeys 0 < λ2 < λ3 = λ4.

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