Category Archives: Quantum Physics

Experts have been warning of something called the quantum apocalypse the point when quantum computers become a reality and render most methods of internet encryption useless.

Boris Johnson promised in November that the UK would go big on quantum computing a new and more powerful way of processing information, based on quantum physics. If you imagine a standard computer to be like a horse and cart, then a quantum computer is more like a sports car a huge leap forward, explained the BBC.

The UK is aiming to secure 50% of the global quantum computing market by 2040, said The Guardian, by investing in the National Quantum Computing Centre in Harwell, Oxfordshire. But the US and China have already taken huge steps to revolutionise research in the field, with the Americans achieving a dramatic lead in quantum computing patents, said Scientific American.

A leaked Google research paper published in 2019 suggested that a computer designed by the tech giant had achieved quantum supremacy defined by The Independent as the ability to perform a calculation that was far beyond the reach of todays most powerful supercomputers.

The paper said that Googles 72-qubit computer took just 200 seconds to perform a calculation that would have taken a supercomputer around 10,000 years to complete.

There is hope that the sophistication of quantum computers could enable scientists to design new chemicals, paving the way for advanced medicines and materials. It could also help weather forecasts and stock trades, and even combat global heating.

Quantum computing gives us a way to model nature better, said Jay Gambetta, a vice-president of quantum computing at IBM, which boasts the worlds most powerful quantum processor.

However, there is also what the BBC has described as a dark side to quantum computing. Current computers would take years, decades and even centuries to crack the encryption codes created by todays machines, but the fact that a quantum computer could theoretically do this in just seconds poses an enormous cybersecurity risk.

The notion of all the worlds most encrypted files from WhatsApp messages to online banking to government data suddenly being broken into thanks to the advent of quantum computing is known as the quantum apocalypse.

The quantum apocalypse could also mark the end of cryptocurrencies like bitcoin, as it would make the blockchain network which is considered to be pretty much hack-proof insecure. UK cybersecurity firm Post Quantum has said that if measures are not put in place, then bitcoin will expire the very day the first quantum computer appears.

The quantum apocalypse isnt a problem that can be left to the next generation to solve. Tim Callan, chief compliance officer at cybersecurity firm Sectigo, warned The Independent that quantum computers could reach the point of defeating our current encryption systems within the next 10 or 15 years.

When that happens, our modern systems of finance, commerce, communication, transportation, manufacturing, energy, government, and healthcare will for all intents and purposes cease to function, he added.

This prognosis was echoed in a BBC interview with Ilyas Khan, chief executive of the Cambridge and Colorado-based company Quantinuum. Quantum computers will render useless most existing methods of encryption, he said. They are a threat to our way of life.

But its not all doom and gloom. As data scientists make advances in the world of quantum computing, theyre also working to create quantum-resistant algorithms to protect our digital footprints.

In the UK, all top secret government data has already been classified as post-quantum, said the BBC. This means that it uses new forms of encryption that scientists believe will standup to quantum computers.

If we werent doing anything to combat [the quantum apocalypse] then bad things would happen, an unnamed Whitehall official told the broadcaster.

None of this comes cheap. The UK has invested millions into the industry over the last few years and that amount is only going to rise. But if you listen to the experts, the consequences of the quantum apocalypse could be so catastrophic that advancing our current systems of encryption is most definitely money well spent.

See the article here:

What is the quantum apocalypse? - The Week UK

image:BECCAL logo view more

Credit: BECCAL

In early December 2021, the project "Development of a laser system for experiments with Bose-Einstein condensates on the International Space Station within the BECCAL payload (BECCAL-II)" commenced, with the involvement of a team of researchers led by Professor Patrick Windpassinger and Dr. Andr Wenzlawski from Johannes Gutenberg University Mainz (JGU). In collaboration with Humboldt-Universitt zu Berlin, the Ferdinand-Braun-Institut (FBH) and Universitt Hamburg, the researchers will develop a laser system for the BECCAL experiment to study ultracold atoms on board the International Space Station (ISS).

The BECCAL experiment is a multi-user platform that will be open to numerous national and international scientists to test their ideas in practice. The platform will enable them to conduct a wide range of experiments in fields such as quantum sensing, quantum information, and quantum optics.

Transport of the BECCAL payload to ISS scheduled for early 2026

The ISS offers a unique combination of weightlessness, accessibility, and a large number of experiments. This will make it possible, among other things, to carry out high-precision experiments such as testing Einsteins equivalence principle. "Ideally, the experiments require the ultracold atom cloud to be completely free of any forces. Weightlessness permits such conditions," said Dr. Andr Wenzlawski from the Windpassinger group at Mainz University.

The BECCAL experiment is a successor to the CAL project, which has conducted numerous experiments aboard the ISS since 2018. BECCAL is intended to enhance the experimental capabilities on board the ISS, especially in the fields of precision atomic interferometry and the manipulation of atoms with detuned optical fields. An additional improvement of the overall performance is being sought by the implementation of new technological approaches to preparing atomic ensembles. The payload is scheduled for launch in early 2026 and will directly replace the CAL apparatus in the ISS Destiny module.

In the subproject, which is funded with EUR 3.4 million, the group led by Professor Patrick Windpassinger from the Institute of Physics at JGU will work together with Universitt Hamburg to develop and realize a Zerodur-based optical splitting and switching system and implement it into the BECCAL payload. These developments will draw on the findings of numerous previous experiments conducted in microgravity conditions, such as MAIUS, QUANTUS, and KALEXUS, in all of which JGU participated. "These experiments have allowed us to lay the technological foundations for running such an extremely complex experiment as well as to perform initial fundamental tests on the feasibility of the envisaged experiments," said Wenzlawski.

The robust laser modules necessary for the experiment are being supplied by the FBH, which is currently manufacturing 55 of the narrow-band laser sources. Humboldt-Universitt zu Berlin is coordinating the integration of these laser modules along with the optical beam splitting and switching benches into a compact overall system. The project is being financed by the German Space Agency of the German Aerospace Center (DLR) with funding from the German Federal Ministry for Economic Affairs and Climate Action, following a resolution by the German Bundestag.

Related links:https://www.qoqi.physik.uni-mainz.de/ Experimental Quantum Optics and Quantum Information research group at the JGU Institute of Physics ;https://www.dlr.de/qt/en/desktopdefault.aspx/tabid-13511/23496_read-54021/ Bose Einstein Condensate and Cold Atom Laboratory (BECCAL) ;https://www.physik.hu-berlin.de/en/qom Optical Metrology group at Humboldt-Universitt zu Berlin ;https://www.fbh-berlin.de/en/research/quantum-technology/quantum-photonic-components/fundamental-physics Integrated quantum technology group at Ferdinand-Braun-Institut gGmbH

Read more:https://www.uni-mainz.de/presse/aktuell/13342_ENG_HTML.php press release "Atom interferometry demonstrated in space for the first time" (13 April 2021) ;https://www.uni-mainz.de/presse/aktuell/6645_ENG_HTML.php press release "Bose-Einstein condensate generated in space for the first time" (31 Oct. 2018) ;https://www.magazin.uni-mainz.de/8106_ENG_HTML.php JGU MAGAZINE: "Pioneering measurements in space" (17 Feb. 2017) ;https://www.uni-mainz.de/presse/aktuell/260_ENG_HTML.php press release "Experiment involving ultracold rubidium lifts off with research rocket" (2 Feb. 2017)

Disclaimer: AAAS and EurekAlert! are not responsible for the accuracy of news releases posted to EurekAlert! by contributing institutions or for the use of any information through the EurekAlert system.

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Collaborative research project on quantum technology starts on the International Space Station - EurekAlert

Future history

The NSAs Research Directorate is descended from the Black Chamber, the first group of civilian codebreakers in the United States who were tasked with spying on cutting-edge technology, like the telegraph. Existing only from 1919 to 1929, the group decoded over 10,000 messages from a dozen nations, according to James Bamfords 2001 book Body of Secrets: Anatomy of the Ultra-Secret National Security Agency. In addition to groundbreaking cryptanalytic work, the group succeeded by securing surveillance help from American cable companies like Western Union that could supply the newly minted US spies with sensitive communications to examine.

The Black Chamber was shut down amid scandal when US Secretary of State Henry Stimson found out the group was spying on American allies as well as foes. The incident foreshadowed the 1975 Church Committee, which investigated surveillance abuses by American intelligence agencies, and the 2013 Snowden leaks, which exposed vast electronic surveillance capabilities that triggered a global reckoning.

Just eight months after the Black Chamber was shuttered, the US, faced with the prospect of crippled spying capabilities in the increasingly unstable world of the 1930s, reformed the effort under the Armys Signals Intelligence Service. One of just three people working with the Black Chambers old records, one of the founders of the SIS, which Bamford reports was kept a secret from the State Department, was the mathematician Solomon Kullback.

Kullback was instrumental in breaking both Japanese and German codes before and during World War II, and he later directed the research and development arm of the newly formed National Security Agency. Within a year, that evolved into the directorate as we know it today: a distinct space for research that is not disrupted by the daily work of the agency.

Its important to have a research organization, even in a mission-driven organization, to be thinking beyond a crisis, says Herrera, though he adds that the directorate does dedicate some of its work to the crisis of the day. It runs a program called scientists on call, which allows NSA mission analysts facing technical challenges while interrogating information to ask for help via email, giving them access to hundreds of scientists.

But the lions share of the directorates work is envisioning the technologies that are generations ahead of what we have today. It operates almost like a small, elite technical college, organized around five academic departmentsmath, physics, cyber, computer science, and electrical engineeringeach staffed with 100 to 200 people.

The cybersecurity department defends the federal governments national security and the countrys military-industrial base. This is the highest-profile department, and deliberately so. Over the last five years, the previously shadowy NSA has become more vocal and active in cybersecurity. It has launched public advisories and research projects that would once have been anathema for an organization whose existence wasnt even acknowledged until 20 years after its founding.

Now the products of NSA research, like Ghidra, a free, sophisticated reverse engineering tool that helps in the technical dissection of hacking tools, as well as other software, are popular, trusted, and in use around the world. They serve as powerful cybersecurity tools, a recruiting pitch, and a public relations play all wrapped into one.

The physics department, which Herrera once directed, runs dozens of laboratories that conduct most of the work on quantum information sciences, but it has a much wider remit than that. As physical limits in the ability to squeeze more transistors into chips threaten to slow and halt 60 years of predictably rapid computing growth, its physicists are exploring new materials and novel computing architectures to drive the next generation of computing into a less predictable future, exactly the kind of task the directorate was given when it first came into existence.

Meanwhile, the electrical engineering department has been looking closely at the physics and engineering of telecommunications networks since the internet first arose. As well as the issues around 5G, it also tackles every facet of the digital world, from undersea cables to satellite communications.

Some prospects on the horizon dont fit neatly into any particular box. The computer science departments work on artificial intelligence and machine learning, for example, cuts across cybersecurity missions and data analysis work with the mathematicians.

Herrera repeatedly raises the prospect of the directorate needing to develop greater capabilities in and understanding of rapidly advancing fields like synthetic biology. The NSA is hardly alone in this: Chinese military leaders have called biotech a priority for national defense.

Much of the competition in the world now is not military, Herrera says. Military competition is accelerating, but there is also dissemination of other technologies, like synthetic biologies, that are frankly alarming. The role of research is to help the NSA understand what the impact of those technologies will be. How much we actually get involved, I dont know, but these are areas we have to keep an eye on.

Read more here:

Meet the NSA spies shaping the future - MIT Technology Review

Today is the release date for the UK edition of A Brief History of Timekeeping (the US edition came out last Tuesday), and Ive been doing a lot of publicity interviews on both sides of the Atlantic. One of the most frequent questions Ive been asked in this process is Whats the most surprising thing you learned in researching this book? and that seems like a decent topic for a publication-day post.

Before I go into the list, though, one important note of background: my training as a professional physicist mostly took place at the National Institute of Standards and Technology (NIST) in Gaithersburg, MD, and the people in the lab next door to mine were literally working on an improved cesium atomic clock. As a result, Im a little jaded when it comes to modern clocks based on quantum physics, just because theyre so familiar to me. That means most of the things that surprised me come from the realm of history.

Anyway, with that out of the way, heres a list of things I found surprising when I began digging into the subject for the course that eventually became the backbone of the book:

IRELAND - NOVEMBER 14: Newgrange Stone Age Passage Tomb (Unesco World Heritage List, 1993), County ... [+] Meath, Ireland. ca 3200 BC. (Photo by DeAgostini/Getty Images)

Precision Timekeeping Is Really Ancient:

I sort of knew this going in, because I always planned to start the story with neolithic solstice markers like Newgrange and Stonehenge. But it was still remarkable to me to learn how long ago a lot of the key elements of astronomical timekeeping were known. The basic idea of making a solstice marker is really simple it just needs a couple of sticks and some patience so its not that surprising that people could manage it thousands of years ago. But lots of other fairly sophisticated bits of science and technology are also thousands of years old. Around 1500 BCE, an Egyptian court official bragged named Amenemhet bragged about inventing a water clock that could keep accurate time through the whole year, which is probably the ancestor of the Karnak clepsydra, whose tapered shape provides a remarkably constant flow rate, and whose seasonal markings reflect a good knowledge of the changing length of the days. The Egyptians were aware of the 1400-year Sothic cycle describing the drift of a 365-day year relative to the seasons (because their civilization lasted long enough to see it, twice), and the Babylonians knew about the Metonic cycle of adding months to keep a lunar calendar in synch with the seasons and the Saros cycle of eclipses for several centuries BCE. Thats a depth of history thats really impressive.

Tablet, Old Babylonian, circa 1800-1600BC. Astronomical Tablet showing the risings and settings of ... [+] Venus. cuneiform script. Dimensions: height: 7.5 cmwidth: 9 cmthickness: 2.8 cmArtist Unknown. (Photo by Ashmolean Museum/Heritage Images/Getty Images)

Some of the Most Important Innovations Are Anonymous:

We can attach names to some really ancient timekeeping discoveries Amenemhet and his clepsydra, Meton of Athens and the cycle he cribbed from the Babylonians but some much more recent inventors remain anonymous. We have no idea who invented the first mechanical clock, for example verge-and-foliot clocks spring up in medieval Europe, and spread rapidly across the continent, but theres no clear record of their invention, or who was responsible. Similarly, we have no idea who made the first sandglass they just start showing up all over the place. There are scattered claims that one person or another invented these things, but most of those werent written down until centuries later, so theyre doubtful at bet.

UNITED KINGDOM - APRIL 18: Oil on canvas painting by Thomas Hudson (1701-1779), showing Graham ... [+] (1673-1751) seated beside a mercury compensating pendulum in an open clock case, c 1710. Following his apprenticeship with London clockmaker Henry Askem, Graham, the inventor of the mercurial pendulum, was considered one of the greatest instrument makers of his day. He made various astronomical instruments, and contributed significantly to the advancement of precision timekeeping. Dimensions (unframed): 1200mm x 960mm. (Photo by SSPL/Getty Images)

Increases in Precision Can Be Astonishingly Rapid:

Prior to the 1600s, few clocks had minute hands because most mechanical clocks werent accurate enough for it to be worthwhile they would require re-setting by checking against the sun on a regular basis. When Galileo Galilei was doing experiments on free fall and pendulum motion, and Tycho Brahe was making the astronomical observations that led to our modern understanding of the solar system, they mostly used water clocks as few if any mechanical clocks of the time were up to the task. The first pendulum clock was built in 1657; within 60 years, George Graham and John Harrison in England were making pendulum clocks that compensated for changes in temperature to such a degree that they were good to around one second a month. We see similarly rapid increases in precision with the introduction of quartz clocks in 1930, and atomic clocks in 1955, and arguably with laser-cooled fountain clocks circa 2000 and optical-frequency clocks in the following decade. These are now accurate enough to measure the gravitational influence on time from an altitude change of centimeters. When scientists and engineers get hold of a good new way to keep time, they turn it into a great way to keep time in a hurry.

[UNVERIFIED CONTENT] Multi-face public clock in a ball shaped brass housing in front of the US flag ... [+] in Grand Central Terminal from low perspective framed with two chandelier

Time Zones Are a Corporate Creation:

In the book, I describe the introduction of time zones in the US as happening via a quinessentially American process: introduced by massive corporations acting to pre-empt legislation. The first national system of standardized time zones came in in 1883, replacing a patchwork of local times based on the sun with a system of broad zones based on the boundaries between rail companies. This was largely the work of William Allen, the Secretary of the General Time Convention of the railroad association, who explicitly wrote that the railroads should adopt his standarization scheme because there is little likelihood of any law being adopted in Washington... that would be as universally acceptable to the railway companies. The companies agreed, and signed on to Allens plan, and lobbied state and local governments to synchronize their clocks with railroad time, rather than the other way around. (I wrote more about this a few months ago here.)

Ambrogio Lorenzetti (Italian, 12851348), Allegory of Good and Bad Government: Good Government, ... [+] fresco, 1338-9, Palazzo Pubblico, Siena, Italy (Photo by VCG Wilson/Corbis via Getty Images)

Sandglasses and Mechanical Clocks Were Invented at the Same Time:

This stands as the single most surprising thing I learned in the process of research for this book. If you look at a sand timer, it seems like an incredibly ancient technology, something that mustve been in use since Egyptian times. In fact, though, the earliest unambiguous reference to a sandglass that we know of is its appearance in a fresco in Siena, Italy, painted in the 1330s. Its just... there in a way that suggests the artist knew it was something the audience would recognize, so they had probably been around for a while, putting the invention sometime in the 1200s. Which is also when the first verge-and-foliot mechanical clocks start popping up in church towers all over Europe.

So, while a sandglass seems like something incredibly old, and ticking mechanical clocks feel relatively modern, theyre actually invented at around the same time. That was really surprising to me, actually the single most surprising fact that I learned in this process. (But again, Im a weirdo physicist who knew a lot about atomic clocks before starting...)

Read the original here:

The Five Biggest Surprises From The History Of Timekeeping - Forbes

About the Centre for Quantum Technologies

The Centre for Quantum Technologies (CQT) is a research centre of excellence in Singapore. It brings together physicists, computer scientists and engineers to do basic research on quantum physics and to build devices based on quantum phenomena. Experts in this new discipline of quantum technologies are applying their discoveries in computing, communications, and sensing.

CQT is hosted by the National University of Singapore and also has staff at Nanyang Technological University. With some 180 researchers and students, it offers a friendly and international work environment.

Learn more about CQT atwww.quantumlah.org

Job Description

The successful candidate will work as part of a collaborative programme between the Centre for Quantum Technologies and Oxford University to develop quantum dot-based quantum light sources for distributed photonic networks for future quantum information processing.

Significant experience of quantum optics or spectroscopy of devices, materials, or nanomaterials is essential and should be accompanied with a demonstrable track record of assembling and operating optical set-ups, lasers, APDs, time-resolved spectroscopy, helium flow cryostats, and control software (e.g. Labview). Knowledge of exciton dynamics and material synthesis is an advantage because the appointee will be expected to contribute to sample fabrication. The appointment is for an initial period of one year, renewable subject to performance.

Job Requirements

More Information

For enquiries and details about the position, please contact Tristan Farrow atcqttf@nus.edu.sg.

Please include your consent by filling in the NUS Personal Data Consent for Job Applicants.

Employment Type : Full-time

Applications can be submitted via the link below and should contain: the latest CV, and letter of recommendation (if any).

Job requisition ID : [[11859]]

Additional Information

At NUS, the health and safety of our staff and students is one of our utmost priorities and COVID-vaccination supports our commitment to ensure the safety of our community and to make NUS as safe and welcoming as possible. Many of our roles require significant amount of physical interactions with student / staff / public members. Even for job roles that can be performed remotely, there will be instances where on-campus presence is required.

With effect from 15 January 2022, based on Singapores legal requirements, unvaccinated workers will not be able work at the NUS premises. As such, we regret to inform that job applicants need to be fully COVID-19 vaccinated for successful employment with NUS.

MOM Updated advisory on COVID-vaccination at the Workplace, subject to changes in accordance with the national COVID-19 measures

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Research Fellow, Quantum Information Processing, Centre For Quantum Technologies job with NATIONAL UNIVERSITY OF SINGAPORE | 279964 - Times Higher...

Kenneth Hicks| The Columbus Dispatch

If you likesuperheromovies, you may have heard about the new one starring Spiderman. I havent seenit, but I heard that it explores a topic called the multiverse.In its simplest form, theidea isthatmultiple universes exist, separate from our own but parallel in time.

Thisideamay sound like fantasy, but actuallyitsbased in science.

The multiverse is often used in cosmology as a way to justify the Big Bang.Our current theory of the Big Bang isfroma process called inflation, where space and time can expand at an exponential rate according to Einsteins general relativity,triggered bya quantum fluctuation in the empty vacuum.

To unpack that last sentence would take a book, so just focus on the last part: it all began with a small quantum fluctuation.

In quantum mechanics, fluctuations in space-time (where particle pairs can pop into existence for a fleeting moment, then disappear) are part of quantum theory. Quantum theory is one of thefoundations of modern physics, having been verified by countless experiments.For example, quantum theoryled to the invention oftransistors, which are used inallcomputers and cellphones.

The quantum fluctuation that led to the Big Bangwas averylow probability event.You might call it a once-in-a-universe likelihood.But if it happened once, then maybe it could have happened twice (or a multitude of times) somewhere in the vastness of the empty void that preceded the Big Bang. Hence, the idea of the multiverse.

The idea of the multiverse has been extended from the vastness of space to the vastness of time.Here, the idea is that whenever a choice is made, the universe splits into different branches in time, spawning two universes, one stemming from each choice.

This idea is key to understanding quantum theory, where probabilistic outcomesdetermine the fate of small particles. This interpretation of quantum theorystemsfrom the work of Nobel Laureate Richard Feynman, who developed the path-integral approach.

The problem with the multiverse theory is that it cant be proven correct ornot.Just like we cant move backward in time, we cantseebeyond the space of our universe.In other words, we cant make contact with any other universe except our own.

Popularmovies would have you think thatthere is a wayfor youtomove backward in time(or into an alternate universe)but thatsfiction.

Although Spiderman isallowedone to move around freely inthe multiverse,we must live in the real world. But if it were possibleto discover a connection to the multiverse, thebeststarting point wouldlikelybe adegreein quantum physics, just like the fictional character of Spidermans alias, Peter Parker.

More here:

Astronomy: The multiverse might exist outside of Marvel - The Columbus Dispatch

Quantum physicists in the city have conducted experiments proving that reality as we think of it may not exist and in the process have not only conclusively disproved an Einsteinian idea of reality but have also paved the way for more secure information transfer.

That all of this should be achieved by quantum scientists should come aslittle surprise. Quantum mechanics has already been expanding our concept of what reality is. Previous experiments around the world, for example, have shown that particles can be in more than one place at a time, but a key tenet of quantum theory is that an object only assumes a definite position if it is seen by the observer.

Bothered by this, Albert Einstein famously said, I like to think that the moon is there even if I am not looking at it.

His statement reflects the every-day believed notion of realism, which suggests that a system has well-defined properties in any instant, even when not measured. But what if the quirks of quantum physics go beyond mere atoms or particles?

Testing realism

This has prompted a spate of experiments to determine, in the words of New Scientist, if there is a hard boundary between the quantum and classical worlds. Central to this is the Leggett-Garg inequality, devised in 1985 by Anthony Leggett and Anupam Garg. This inequality looks for correlations between measurements to see whether quantum or classical rules are being followed, the New Scientist states. In essence, it is a means of testing realism.

Professor Urbasi Sinha of Raman Research Institute (RRI) explained that the experimental violation of such inequality would not only falsify realism but also would confirm that quantum mechanics is not limited to the micro-world, but can be applied to bigger objects, such as the moon.

Leggett and Garg realized they could test the quantumness of big objects in theory. Their inequality could tell us whether realism holds true in the everyday world, she said.

She added that, this could also make way for harnessing non-classicality or quantumness of single photons for technological applications such as secure quantum communications and quantum sensing, which are crucial in todays requirements of secure information transfer.

In recent years, Leggett-Garg experiments carried out on various quantum systems from superconducting fluids and photons to atomic nuclei and tiny crystals have demonstrated that the microscopic world is non-real. For this they have found ways of measuring particles without disturbing it.

Testing the macroscopic limit of quantum mechanics is an important area of research because it can reveal up to what extent quantum principles dominate revealing the quantum-classical boundary.

However, these experiments have limitations. Scientists worldwide are trying to come up with better technology and appropriately designed strategies for achieving a fully conclusive experimental test.

Now, a team of scientists from RRI has successfully addressed this challenge.

In the course of a two-year experiment, she showed a significant amount of violation of Leggett Garg inequality by studying single photons.

The experiment was performed at the Quantum Information and Computing laboratory of RRI and was led by Urbasi along with her PhD student Kaushik Joarder. Theoretical contributions from Professor Dipankar Home of the Bose Institute Kolkata and Dr Debashis Saha of the S N Bose Centre for Basic Sciences Kolkata played a significant role in the work.

First experiment

The work, published in PRX Quantum, is the first ambiguity-free experiment to show violation of Leggett Garg inequalities.

The team conducted the experiment with single photons (particles of light) and proved the quantumness of the single photon comprehensively. This is the first experiment that shows the most decisive refutation of the notion of realism by the closure of what are known as loopholes plaguing all relevant experiments to date, Urbasi said.

Loopholes are elements, such as equipment limitations or study-related factors which can inadvertently alter the experiment or conspire to deviate results, she told DH.

She added that the strategies and technologies developed for the closure of all the existing loopholes will prove to be very useful for harnessing such non-classicality/quantumness of single photons for technological applications in secure quantum communications and quantum sensing.

Moreover, the experiment further shows remarkable agreement with quantum physics predictions. In our analysis, we have been able to show that not only are we violating the Leggett Garg inequality in a loophole-free manner, but that we were also showing remarkable agreement with the predictions of quantum mechanics, Urbasi said.

This work was partially funded by the Centre of Excellence in quantum technologies grant from the Ministry of Electronics and Information Technology as well as the Quantum Enabled Science and Technology grants from DST.

Here is the original post:

An experiment that disproves Einstein's idea of reality - Deccan Herald

The grandfather paradox is a self-contradictory situation that arises in some time travel scenarios that is illustrated by the impossible scenario in which a person travels back in time only to kill their grandfather (who could no longer go on to produce one's parent, and hence where does that leave you and your ancestor-killing event?). The paradox is sometimes taken as an argument against the logical possibility of traveling backward in time, according to the Stanford Encyclopedia of Philosophy. Within the framework of modern physics, however, there are ways to avoid the paradox without dispensing with time travel altogether.

Related: 5 sci-fi concepts that are possible (in theory)

Let's suppose you have a time machine that allows you to travel back into the past. While you're there, you accidentally kill one of your grandparents or any other direct ancestor before they have any offspring. That would alter a whole chain of future events, including your own birth, which would no longer happen. But if you weren't born in the future, then you couldn't kill your ancestor in the past hence the paradox. It's a scenario that became popular in the science-fiction magazines of the 1920s and 1930s, according to the Historical Dictionary of Science Fiction, and the name "grandfather paradox" was firmly established by 1950.

Actually, you don't even need to kill anyone; there are many other ways you could change history that would result in your future non-existence. Perhaps the best known example is the movie "Back to the Future," in which the time-traveling protagonist inadvertently drives a wedge between his parents before they were married and then has to work frantically to bring them together again.

Moving from science fiction to science fact, one person who was eminently qualified to talk about the realities of time travel was the late Stephen Hawking, arguably the most brilliant physicist of recent times. In 1999, he gave a lecture on "space and time warps," which showed how Einstein's theory of general relativity might make time travel possible, by bending space-time back on itself.

One theoretical possibility that would allow time travel (and thus the ability to somehow kill off a critical ancestor) is a special kind of wormhole. Among the most dramatic consequences of general relativity, wormholes are often described as shortcuts between one point in space and another. But, as Hawking explained in his lecture, a wormhole could possibly loop back to an earlier point in time a situation technically known as a "closed time-like curve" (CTC).

But if physics allows backward time travel, wouldn't the grandfather paradox still cause issues? Hawking suggested two possible ways to get around the paradox in this scenario. First, there's what he referred to as the "consistent histories" model, in which the whole of time past, present and future is rigidly predetermined; in that way, you can only travel back to an earlier point in time if you had already been there in your own history. In this "block universe" model, as it's sometimes called, one could travel to the past but doing so would not alter it, according to the Australian Broadcasting Corporation. Taking this view, the grandfather paradox could never arise. With Hawking's second option, on the other hand, the situation is more subtle.

This second approach to traveling back in time invokes quantum physics, where an event may have several possible outcomes with different likelihoods of occurring.

As described by the Stanford Encyclopedia of Philosophy, the "many worlds" interpretation of quantum theory sees all these various outcomes as occurring in different, "parallel" timelines. In this view, the grandfather paradox could be resolved if the time traveler starts out in a timeline where their grandfather lived long enough to have children, and then after going back and killing their forebear continue along a parallel time track in which they will never be born. (Stanford Encyclopedia has a more detailed look at why you cant jump back and forth between parallel timelines at will.) As Hawking pointed out in his 1999 lecture, this seems to be the implicit assumption behind sci-fi treatments such as "Back to the Future."

At the time that movie was made in 1985, the "parallel world" explanation of the grandfather paradox was merely a philosophical conjecture. In 1991, however, it was put on firmer ground by the physicist David Deutsch, as New Scientist reported at the time. Deutsch showed that, while parallel timelines are normally incapable of interacting with each other, the situation changes in the vicinity of a closed time-like curve (CTC), when a wormhole curves back on itself. Here, just as the sci-fi writers imagined, the different timelines are able to cross over so that when a CTC loops back into the past, it's the past of a different timeline. If that's proven, then you really could kill an infant grandparent without paradoxically eliminating yourself in the process. In that case, your grandfather would never have existed only in one parallel world. And you, the grandfather-killer, would only have existed in the other.

As surprising as it sounds, there's actually some experimental support for Deutsch's solution to the grandfather paradox. In 2014, a team at the University of Queensland examined a simpler time-travel scenario that entailed a similar logical paradox. The researchers described the work in their paper published that year in the journal Nature Communications. The idea was that a subatomic particle had to go back in time to flip the switch that resulted in its creation; if the switch wasn't flipped, the particle would never exist in the first place.

A key feature of Deutsch's theory is that the various probabilities have to be self-consistent. For instance, in the Queensland research example, if there's a 50:50 chance the particle travels back in time, then there must also be a 50:50 chance that the switch gets flipped to create that particle in the first place. In the absence of a time machine, the researchers set up an experiment involving a pair of photons, which they claimed was logically equivalent to a single photon traveling back in time to "create" itself. The experiment was a success, with the results validating Deutsch's self-consistency theory.

Killing your grandfather when he was a child is a sure-fire way to ensure you're never born. But there are also subtler possibilities for messing up the timeline. In a sufficiently complex system, even the tiniest change can have serious long-term consequences as in the butterfly effect, by which the flapping of a butterfly's wings can eventually trigger a tornado thousands of miles away. Sci-fi writer Ray Bradbury produced a time travel counterpart to this in his 1952 story "A Sound of Thunder," which can be read online at the Internet Archive. Bradbury's protagonist travels back to the time of the dinosaurs, where he accidentally steps on a butterfly then returns to the present to find society changed beyond recognition. It's easy to imagine that, if the societal changes were sweeping enough, the time traveler might have prevented his own birth as surely as if he'd slain a grandparent.

But would that really be the case, using the quantum approach to the grandfather paradox? Recent work at the Los Alamos National Laboratory indicates that the course of history is more resilient than the butterfly effect might suggest. The researchers used a quantum computer to simulate time travel into the past, where a piece of information was deliberately damaged the computational equivalent of stepping on a Jurassic-era butterfly. But unlike Bradbury's story, the knock-on effect in the "present" of the computer simulation turned out to be relatively small and insignificant. That, of course, is great news for would-be time travelers. As long as you refrain from blatantly silly acts like killing a direct ancestor, it may be possible to go back in time without any paradoxical consequences at all.

Historical Dictionary of Science Fiction. https://sfdictionary.com/view/2178/grandfather-paradox

"Many-Worlds Interpretation of Quantum Mechanics," Stanford Encyclopedia of Philosophy, 2021. https://plato.stanford.edu/entries/qm-manyworlds/

"Time Travel without the Paradoxes," New Scientist, 1992. https://www.newscientist.com/article/mg13318143-000-science-time-travel-without-the-paradoxes/

"The block universe theory, where time travel is possible but time passing is an illusion," Australian Broadcasting Corporation, 2018. https://www.abc.net.au/news/science/2018-09-02/block-universe-theory-time-past-present-future-travel/10178386

"Experimental simulation of closed timelike curves," Nature Communications, 2014. https://www.nature.com/articles/ncomms5145

"A Sound of Thunder," Ray Bradbury, Internet Archive. https://archive.org/details/Planet_Stories_v06n04_1954-01/page/n5/mode/2up

"Simulating quantum 'time travel' disproves butterfly effect in quantum realm," Los Alamos National Laboratory, 2020. https://www.lanl.gov/discover/news-release-archive/2020/July/0728-quantum-time-travel.php

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What is the grandfather paradox? - Livescience.com

Physicists hope to detect asymmetry in spinning black holes using NASA's LISA telescope to finally provide proof of quantum gravity.

Detecting gravitational waves using Earth-based observatories has become a powerful tool for studying the properties of black holes.

Recently, a group of theoretical physicists has analyzed the process of gravitational wave emission and showed that the proposed space-based gravitational wave detector, LISA, will have enough sensitivity not only to detect more black hole mergers, but to measure a feature of spinning black holes an asymmetry between their northern and southern hemispheres in particular that will help elucidate a longstanding mystery.

The asymmetry is imprinted in a precise waveform of the gravitational radiation and is absent in the theory of general relativity, the most accepted theory of gravity, but is predicted by its quantum extension.

Quantum mechanics has been successfully used to study and explain the behavior of atoms, nuclei, and subatomic particles, while general relativity, which describes gravity as a curvature of spacetime, explains and predicts the dynamics of stars, galaxies, and the universe as a whole.

The unification of quantum mechanics and general relativity has been a formidable task made difficult by the fact that the usual rules of quantization required to convert a classical theory into a quantum one dont appear to work for gravity.

Since the typical scale of physical systems described by these two theories differ by many orders of magnitude, there is no need to use both simultaneously to describe a given event or behavior. But sometimes this is necessary. For example, a quantum mechanical description of gravity is needed to understand what happens in the vicinity of black holes surfaces and centers, as well as describe the behavior of the universe during the first moments of its life.

So far, theoretical physicists have proposed a number of theories for quantum gravity, but experimental studies are extremely complicated. The problem being that the typical energy scale at which the quantum gravity effects become important in the elementary particles interactions which scientists usually study to understand the fundamental physics are many orders of magnitude higher than the energies that can be achieved in colliders. For this reason, researchers have been seeking another way to explore it.

The most promising systems in which these effects are possible to measure are black holes, and physicists have already begun probing quantum effects in their physics by studying their membranes, which should exist above a black holes surface according to some theories of quantum gravity.

This is done by analyzing gravitational waves emitted during black hole merger events, which physicists currently detect with the Earth-based gravitational wave observatories LIGO and Virgo. Gravitational waves, which were first theorized by Albert Einstein, are ripples in the fabric of spacetime, and black hole mergers are so important in this realm because they generate the most powerful gravitational radiation in the universe.

Recently, two scientists from the Institute for Theoretical Physics in KU Leuven have proposed that studying these waves can also be used to measure the asymmetry between the northern and southern hemispheres of spinning black holes, giving us another chance to study quantum gravity with gravitational waves.

The best way to identify this, the researchers say, is to analyze the amplitude and spectrum of gravitational waves emitted when a relatively light black hole with a mass just a few times larger than the mass of our Sun is consumed by a supermassive neighbor a process that physicists call extreme mass-ration inspirals. The gravitational radiation generated in such events should carry an imprint of the aforementioned black hole asymmetry.

After analyzing gravitational waves from said event, the research team, unfortunately, had to conclude that the change in the gravitational wave signal due to the black hole asymmetry is too small to be detectable by currently operational gravitational wave detectors on Earth.

This problem could be solved with the launch of NASAs space-based gravitational wave observatory, LISA, which will have a much greater sensitivity to tiny changes in the spacetime geometry caused by passing gravitational waves.

LISA consists of three spacecrafts arranged in an equilateral triangle with sides that are 2.5 million km long (in comparison to 4 km arm lengths of LIGO), and moves along an Earth-like orbit around the Sun. When spacetime is distorted by celestial bodies found in our Solar System, the distances between the spacecraft stay the same. But when a gravitational wave passes through the LISA orbit, it leads to small oscillations in the triangle side lengths.

In order to detect these oscillations, laser beams are set up to travel between each spacecraft. When the distances along which the light beams travel change, a pattern in the combined beam signal changes as well, signaling the detection of the wave. The amplitude of the distance change between the spacecraft is proportional to the distance itself. The giant size of the space observatory will allow it to detect very small oscillations in the spacetime geometry, making it extremely sensitive. To put into perspective, LISA is expected to be able to measure relative shifts in the position of each spacecraft at distances that are less than the diameter of a hydrogen atom.

Its launch scheduled for the mid-2030s, and hopefully, it will allow us to put our theories of quantum gravity to the test.

Reference: (preprint) Kwinten Fransen, et al., On Detecting Equatorial Symmetry Breaking with LISA, arXiv:2201.03569

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Black hole asymmetry puts quantum gravity to the test - Advanced Science News

Basics of the general CK oscillator with nonextensivity

Various physical systems subjected to a friction-like force which is a linear function of velocity can be modeled by the formal CK oscillator. The Hamiltonian of the CK oscillator is given by34,35

$$begin{aligned} {hat{H}} = e^{-gamma t} frac{{hat{p}}^2}{2m} + frac{1}{2} e^{gamma t} m omega ^2 {hat{x}}^2, end{aligned}$$

(1)

where (gamma) is a damping constant. This Hamiltonian can be generalized by replacing the ordinary exponential function with a deformed one that is defined by1,37

$$begin{aligned} exp _q {(y)} = [1+(1-q)y]^{1/(1-q)}, end{aligned}$$

(2)

with an auxiliary condition

$$begin{aligned} 1+(1-q)y ge 0, end{aligned}$$

(3)

where q is a parameter indicating the degree of nonextensivity. This generalized function is known as the q-exponential and has its own merit in describing non-idealized dynamical systems. The characteristic behavior of the q-exponential function is shown in Fig.1. In the field of thermostatistics, a generalization of the Gaussian distribution through the q-exponential is known as the Tsallis distribution that is well fitted to many physical systems of which behavior does not follow the usual BG statistical mechanics38.

q-exponential function for several different values of q.

In terms of Eq.(2), we can express the generalized CK Hamiltonian in the form1

$$begin{aligned} {hat{H}}_q = frac{{hat{p}}^2}{2m exp _q{(gamma t)}} + frac{1}{2} exp _q{(gamma t)} m omega ^2 {hat{x}}^2. end{aligned}$$

(4)

This Hamiltonian is Hermitian and, in the case of (q rightarrow 1), it recovers to the ordinary CK one that is given in Eq.(1). From the use of the Hamiltons equations in one dimension, we can derive the classical equation of motion that corresponds to Eq.(4) as

$$begin{aligned} ddot{x} + frac{gamma }{1+(1-q)gamma t}{dot{x}} + omega ^2 x = 0. end{aligned}$$

(5)

In an extreme case where (q rightarrow 0), Eq.(2) reduces to a linear function (1+y). Along with this, Eq.(5) reduces to

$$begin{aligned} ddot{x} + frac{gamma }{1+gamma t}{dot{x}} + omega ^2 x = 0. end{aligned}$$

(6)

If we think from the pure mathematical point of view, it is also possible to consider even the case that q is smaller than zero based on the condition given in Eq.(3). However, in most actual nonextensive systems along this line, the value of q may not deviate too much from unity which is its standard value. So we will restrain to treating such extreme cases throughout this research.

In general, for time-dependent Hamiltonian systems, the energy operator is not always the same as the given Hamiltonian. The role of the Hamiltonian in this case is restricted: It plays only the role of a generator for the related classical equation of motion. From fundamental Hamiltonian dynamics, we can see that the energy operator of the generalized damped harmonic oscillator is given by26,39

$$begin{aligned} {hat{E}}_{q} = {hat{H}}_q/exp _q{(gamma t)}. end{aligned}$$

(7)

Let us denote two linearly independent homogeneous real solutions of Eq.(5) as (s_1(t)) and (s_2(t)). Then, from a minor mathematical evaluation, we have40,41

$$begin{aligned} s_1(t)= & {} {s}_{0,1}sqrt{frac{pi omega }{2gamma (1-q)}} [exp _q{(gamma t)}]^{-q/2} J_nu left( frac{omega }{(1-q)gamma } + omega t right) , end{aligned}$$

(8)

$$begin{aligned} s_2(t)= & {} {s}_{0,2}sqrt{frac{pi omega }{2gamma (1-q)}} [exp _q{(gamma t)}]^{-q/2} N_nu left( frac{omega }{(1-q)gamma } + omega t right) , end{aligned}$$

(9)

where (J_nu) and (N_nu) are the Bessel functions of the first and second kind, ({s}_{0,1}) and ({s}_{0,2}) are constants which have dimension of position, and (nu = {q}/{[2(1-q)]}). From Fig.2, we see that the phases in the time evolutions of (s_1(t)) and (s_2(t)) are different depending on the value of q. Now we can represent the general solution of Eq.(5) in the form

$$begin{aligned} x(t) = c_1 s_1(t) + c_2 s_2(t), end{aligned}$$

(10)

where (c_1) and (c_2) are arbitrary real constants.

Time evolution of (s_1(t)) (A) and (s_2(t)) (B) for several different values of q. We used (omega =1), (gamma =0.1), and (s_{0,1}=s_{0,2}=0.1).

We introduce another time function s(t) that will be used later as

$$begin{aligned} s(t) = sqrt{s_1^2(t)+s_2^2(t)}. end{aligned}$$

(11)

This satisfies the differential equation42

$$begin{aligned} ddot{s}(t) + frac{gamma }{1+(1-q)gamma t}{dot{s}}(t) + omega ^2 s(t) - frac{Omega ^2}{[mexp _q{(gamma t)}]^2} frac{1}{s^3(t)} = 0, end{aligned}$$

(12)

where (Omega) is a time-constant which is of the form

$$begin{aligned} Omega = m exp _q{(gamma t)} [s_1 {dot{s}}_2 - {dot{s}}_1 s_2 ]. end{aligned}$$

(13)

By differentiating Eq.(13) with respect to time directly, we can readily confirm that (Omega) does not vary in time.

In accordance with the invariant operator theory, the invariant operator must satisfy the Liouville-von Neumann equation which is

$$begin{aligned} frac{d {hat{I}}}{d t} = frac{partial {hat{I}}}{partial t} + frac{1}{ihbar } [{hat{I}},{hat{H}}_q] = 0. end{aligned}$$

(14)

A straightforward evaluation after substituting Eq.(4) into the above equation leads to24,40

$$begin{aligned} {hat{I}} = hbar Omega left( {hat{b}}^dagger {hat{b}} + frac{1}{2}right) , end{aligned}$$

(15)

where ({hat{b}}) is a destruction operator defined as

$$begin{aligned} {hat{b}} = sqrt{frac{1}{2hbar Omega }} left[ left( frac{Omega }{s(t)} -i m exp _q{(gamma t)} {dot{s}}(t) right) {hat{x}} + i s(t) {hat{p}} right] , end{aligned}$$

(16)

whereas its hermitian adjoint ({hat{b}}^dagger) is a creation operator. If we take the limit (gamma rightarrow 0), Eq.(16) reduces to that of the simple harmonic oscillator. One can easily check that the boson commutation relation for ladder operators holds in this case: ([{hat{b}},{hat{b}}^dagger ]=1). This consequence enables us to derive the eigenstates of ({hat{I}}) in a conventional way.

The zero-point eigenstate (| 0 rangle) is obtained from ({hat{b}}| 0 rangle =0). The excited eigenstates (| n rangle) are also evaluated by acting ({hat{b}}^dagger) into (| 0 rangle) n times. The Fock state wave functions (| psi _n rangle) that satisfy the Schrdinger equation are different from the eigenstates of ({hat{I}}) by only minor phase factors which can be obtained from basic relations24. However, we are interested in the SU(1,1) coherent states rather than the Fock states in the present work.

The SU(1,1) generators are defined in terms of ladder operators, such that

$$begin{aligned} hat{{mathcal {K}}}_0= & {} frac{1}{2} left( {hat{b}}^dagger {hat{b}} + frac{1}{2}right) , end{aligned}$$

(17)

$$begin{aligned} hat{{mathcal {K}}}_+= & {} frac{1}{2} ({hat{b}}^dagger )^2, end{aligned}$$

(18)

$$begin{aligned} hat{{mathcal {K}}}_-= & {} frac{1}{2} {hat{b}}^2. end{aligned}$$

(19)

From the inverse representation of Eq.(16) together with its hermitian adjoint ({hat{b}}^dagger), we can express ({hat{x}}) and ({hat{p}}) in terms of ({hat{b}}) and ({hat{b}}^dagger). By combining the resultant expressions with Eqs.(17)(19), we can also represent the canonical variables in terms of SU(1,1) generators as

$$begin{aligned} {hat{x}}^2= & {} frac{hbar s^2}{Omega } (2hat{{mathcal {K}}}_0 + hat{{mathcal {K}}}_+ + hat{{mathcal {K}}}_-), end{aligned}$$

(20)

$$begin{aligned} {hat{p}}^2= & {} frac{hbar }{s^2} Bigg [ 2 left( Omega + frac{[mexp _q(gamma t)]^2}{ Omega } s^2{dot{s}}^2 right) hat{{mathcal {K}}}_0 -left( sqrt{Omega } - frac{imexp _q(gamma t)}{ sqrt{Omega }} s{dot{s}} right) ^2 hat{{mathcal {K}}}_+ nonumber \&-left( sqrt{Omega } + frac{imexp _q(gamma t)}{ sqrt{Omega }}s{dot{s}} right) ^2 hat{{mathcal {K}}}_- Bigg ]. end{aligned}$$

(21)

The substitution of the above equations into Eq.(4) leads to

$$begin{aligned} {hat{H}}_q = delta _0(t) hat{{mathcal {K}}}_0 + delta (t) hat{{mathcal {K}}}_+ + delta ^*(t) hat{{mathcal {K}}}_- , end{aligned}$$

(22)

where

$$begin{aligned} delta _0(t)= & {} frac{hbar }{s^2} left( frac{Omega }{mexp _q{(gamma t)}} + frac{1}{Omega } mexp _q{(gamma t)} s^2 {dot{s}}^2 right) + frac{hbar }{Omega } mexp _q{(gamma t)}omega ^2 s^2 , end{aligned}$$

(23)

$$begin{aligned} delta (t)= & {} - frac{hbar }{2 mexp _q{(gamma t)} s^2} left( sqrt{Omega } - i frac{mexp _q{(gamma t)}s{dot{s}}}{sqrt{Omega }} right) ^2 + frac{hbar }{2Omega } mexp _q{(gamma t)} omega ^2 s^2 . end{aligned}$$

(24)

In accordance with Gerrys work (see Ref. 43), Eq.(22) belongs to a class of general Hamiltonian that preserves an arbitrary initial coherent state. In the next section, we will analyze the properties of nonextensivity associated with the SU(1,1) coherent states using the Hamiltonian in Eq.(22).

The SU(1,1) coherent states for the quantum harmonic oscillator belong to a dynamical group whose description is based on SU(1,1) Lie algebraic formulation. The analytical representation of the SU(1,1) coherent states provides a natural description of quantum and classical correspondence which has an important meaning in theoretical physics. On the experimental side, optical interferometers like radio interferometers that use four-wave mixers as a protocol for improving measurement accuracy are characterized through the SU(1,1) Lie algebra44,45.

According to the development of Perelomov46, the SU(1,1) coherent states are defined by

$$begin{aligned} | {tilde{xi }};k rangle = hat{{mathcal {D}}}(beta )|{{{tilde{0}}}} rangle _k , end{aligned}$$

(25)

where (hat{{mathcal {D}}}(beta )) is the displacement operator, (|{{{tilde{0}}}} rangle _k) is the vacuum state in the damped harmonic oscillator, and k is the Bargmann index of which allowed values are 1/4 and 3/4. The basis for the unitary space is a set of even boson number for (k=1/4), whereas it is a set of odd boson number for (k=3/4). Here, the displacement operator is given by

$$begin{aligned} {hat{D}}(beta )= & {} exp left[ frac{1}{2} (beta ^2 hat{{mathcal {K}}}_+ - beta ^{*2} hat{{mathcal {K}}}_-) right] nonumber \= & {} e^{{tilde{xi }} hat{{mathcal {K}}}_+} exp {-2ln [cosh (|beta |^2/2)] hat{{mathcal {K}}}_0} e^{-{tilde{xi }}^* hat{{mathcal {K}}}_-}, end{aligned}$$

(26)

where (beta) is the eigenvalue of ({hat{b}}) and ({tilde{xi }}) is an SU(1,1) coherent state parameter of the form

$$begin{aligned} {tilde{xi }} = frac{beta ^2}{|beta |^2} tanh (|beta |^2/2). end{aligned}$$

(27)

The above equation means that (|{tilde{xi }}| <1). For (k=3/4) among the two allowed values, the resolution of the identity in Hilbert space is given by47

$$begin{aligned} int dmu ({tilde{xi }};k) | {tilde{xi }} ; k rangle langle {tilde{xi }} ; k| = mathbf{1}, end{aligned}$$

(28)

where (dmu ({tilde{xi }};k)=[(2k-1)/pi ] d^2 {tilde{xi }} /(1-|{tilde{xi }}|^2)^2). More generally speaking, this resolution is valid for (k>1/2). For a general case where k is an arbitrary value, the exact resolution is unknown. Brif et al., on one hand, proposed a resolution of the identity with a weak concept in this context, which can be applicable to both cases of (k>1/2) and (k<1/2)47. In what follows, various characteristics of the damped harmonic oscillator with and without deformation in quantum physics, such as quantum correlation, phase coherence, and squeezing effect, can be explained by means of the SU(1,1) Lie algebra and the coherent states associated with this algebra48,49.

The expectation values of SU(1,1) generators in the states (| {tilde{xi }};k rangle) are50

$$begin{aligned} langle {tilde{xi }} ;k | hat{{mathcal {K}}}_0|{tilde{xi }};krangle= & {} k frac{1+|{tilde{xi }}|^2}{1-|{tilde{xi }}|^2 } , end{aligned}$$

(29)

$$begin{aligned} langle {tilde{xi }} ;k | hat{{mathcal {K}}}_+|{tilde{xi }};krangle= & {} frac{2k{tilde{xi }}^*}{1-|{tilde{xi }}|^2 } , end{aligned}$$

(30)

$$begin{aligned} langle {tilde{xi }} ;k | hat{{mathcal {K}}}_-|{tilde{xi }};krangle= & {} frac{2k{tilde{xi }}}{1-|{tilde{xi }}|^2 }. end{aligned}$$

(31)

Using the above equations, the expectation values of the Hamiltonian given in Eq.(22) are easily identified as50,51

$$begin{aligned} {{mathcal {H}}}_{q,k}= & {} langle {tilde{xi }} ;k | {hat{H}}_q |{tilde{xi }};k rangle nonumber \= & {} frac{k}{1-|{tilde{xi }}|^2} { delta _0(t)(1+|{tilde{xi }}|^2) +2 [delta (t){tilde{xi }}^*+delta ^*(t){tilde{xi }}] } . end{aligned}$$

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Analysis of the effects of nonextensivity for a generalized dissipative system in the SU(1,1) coherent states | Scientific Reports - Nature.com