Monthly Archives: July 2013

The Zeno Paradox

The Zeno Paradox

For fun this week we have a short story from Dag on The Zeno Paradox.

A side street in Tokyo. Neon lights in heavy rain. A shady bar with a barman who never speaks unless you don’t pay for your booze. A lonely guy sits in the darkest corner of the bar with a half empty bottle of Yamazaki. Cigarette smoke slithers around his unshaven face, eyes focus on some memories swirling in the dark behind the window. This is the place where men come to absolve their sins before disappearing into the night.

The bar door swings open. A man in a trench coat steps in, pauses to look around. His long shadow stretches towards the lonely guy as if trying to tighten its icy fingers around his throat. The barman gives the newcomer a quick glance only to get back to his world of endless nights when time stands still like the rows of bottles behind him.

“I hate rain”, he mutters to himself.

The newcomer sits in front of the lonely guy.

“William?”. The guy takes a long drag of his cigarette, savours the smoke for awhile, turns his head towards the newcomer and exhales straight into his face.

“Who’s asking?”, he says.

“Wilson. Do you have it?”

William doesn’t reply immediately. He pours himself a glass of whiskey, double shot, looks through it at Wilson, puts it on the table, adds more and then gulps it down like it is his last.

“Yeah,” says William, inhaling the cigarette.”I have it,” he adds, exhaling a thin streak of smoke.

“Give it to me.” Wilson’s voice sounds greedy. William looks straight into his eyes and says almost caringly,

“I’ll give it to you but you must listen to my story first.” “Keep it short, pal,” replies Wilson.

“I loved Gail more than anything, more than myself. I first saw her in a small dancing studio at night. It was raining like today.” William’s voice becomes shaky. He takes another shot of whiskey.

“She was practicing some moves in front of a big mirror. She looked so beautiful, like out of this world. Her body moved across the dance floor with a grace I’d never seen before. I was standing there, glued to that big window and I knew that Gail was the woman I wanted to be with.” He grabs Wilson by the arm and says feverishly “Can you understand that? Can you?!”. Wilson shakes off the hand.

“Take it easy, man” he says dryly.

“We were like Bonnie and Clyde. Lovers, friends. It was a blast but nothing good lasts for long in this twisted universe. Gail fell terminally ill.” William stops, lights another cigarette. Smoke seems to make it easier for him.

“I couldn’t watch her body wasting away”, he pauses, eyes fixated on the swirling cigarette smoke.

“Have you heard about the Zeno paradox?” “No” replies Wilson.

“Zeno claimed that nothing moves because to get from A to B you need to cover half the distance, then the half of the half and so on. Every half requires a finite time to travel but there are infinitely many of them so you won’t cover the distance in a finite time.”

“Nonsense” says Wilson.

“Yeah…. Infinitely many pieces can give you a finite thing”, William pauses, “Not in the quantum world.”

“What do you mean?” William gets Wilson’s attention.

“In the quantum world there is no reality. Observation creates it and this means you can manipulate reality by simply looking at physical systems” William puts out the cigarette. “If you observe them frequently enough you can freeze them forever.”

“That’s how the machine works?” interrupts Wilson.

“Yeah, something like that.”

“Where are the blueprints?” Wilson’s eyes flicker with greed.

“I haven’t finished yet.” William lights another cigarette. “I thought I could keep Gail in a state of suspension until they found a cure.”

“And…?”

“I asked her to dance for me one more time and…” he swallows tears.

“What?” asks Wilson impatiently, pouring William another drink. William ignores it.

“Then I set this… machine… in motion.” William’s voice quivers again. He gulps down the glass of whiskey and goes motionless like a mechanical toy with a discharged battery.

“And?” Wilson prompts him.

“At first it worked beautifully. Gail’s body froze in time… She looked so beautiful.”

“And?!” asks Wilson’s impatiently.

“A few days later I noticed some small changes in her face. Blemishes.” He pauses. “The blemishes started to become fuzzy and larger, slowly transforming Gail’s body into… into…” William swallows hard, his Adam’s apple forcing its way up and down like a piston of a worn out engine, “into something undefined, smeared in space.” William’s hand wipes some invisible grease off his face.

“Couldn’t you stop the machine?” interrupts Wilson.

“It was too late. I would have had to reverse the whole time evolution but I didn’t have enough computational power.” William takes out a notebook. “Here’s the blueprint for the machine.” He throws it on the table. “Can I go now?”

“Where is she now? I mean Gail” asks Wilson ignoring William’s question.

“I’d like to believe that she’s become entangled with the rest of the universe” he pauses, looks into the night behind the window. “And that one day I’ll be able to bring her back, see her dancing again…”

Wilson picks up the blueprint and puts it into an internal pocket of his trench coat.”You know I can’t let you go. We need your expertise. Without you it would take us too long to build the machine.” Wilson wraps his fingers around William’s arm. “Just don’t do anything stupid.”

William looks at Wilson and smiles, his eyes hidden in the shadow.

A side street in Tokyo. Neon lights in heavy rain. A shady bar with a barman who never speaks unless you don’t pay for your booze. A lonely guy sits in the darkest corner of the bar with a half empty bottle of Yamazaki. Cigarette smoke slithers around his unshaven face. A fuzzy, slowly expanding blemish appears at the corner of his eye.

Check out more at Quantum Shorts 2013: Zeno Paradox 

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When knowing a little maths helps.

Most quantum optics textbooks will spend a chapter or two on the theory of open quantum systems. The typical exposition consists of deriving the master equation in the Born-Markov approximation. This is a complicated integro-differential equation governing the evolution of the system state and further simplification requires the specification a concrete model Hamiltonian. The textbook will usually proceed to
give example Hamiltonians and derives the explicit form of the Born-Markov master equation. If you’ve had any exposure to open systems theory from a quantum optician’s perspective then you’ll know that all this is pretty standard. However, just because
it’s in a textbook (or “standard”) doesn’t mean it’s easy to understand. The Born-Markov approximation is rather abstract and explanations of it are often brief. Thus it leaves the reader a lot to swallow, or just simply stuck only after a couple of pages into the
chapter on open systems. Aside from getting the physical principles behind the approximation, the most difficult thing to swallow is that after making the approximation one is still left with a very complicated equation which is supposed to describe valid Markovian evolution for the system, and this is not at all obvious. Just seeing the Born-Markov master equation should make one lose heart that it really works. This is when knowing a bit of maths helps. We fight the abstract with the abstract. I’m referring to what is known as the Lindblad theorem which too often gets swept under the carpet in quantum optics texts. In brief this theorem states what the most general form of a Markovian master equation must look like, often called the Lindblad form*. Knowing this, the natural question is then whether the Born-Markov master equation leads to master equations in the Lindblad form? Lo and behold it does! So it appears then that one should just learn Lindblad’s theory and ditch Born and Markov since Lindblad’s result is much more general. Well, no, because Lindblad’s theorem is too general, mostly just formal mathematics. Lindblad’s theorem tells us what the correct answer has to look like but it doesn’t tell us how to get to the answer i.e. Lindblad’s theorem doesn’t provide a recipe for how to derive a Markovian master equation from model Hamiltonians (which is what physicists want). Learning Lindblad’s result does however settle any doubt that one had about the Born-Markov approximation when it was first encountered and this is great. Furthermore, it leads to a greater appreciation of the Born-Markov approximation for the reason that it actually points out the physics leading** to valid Markovian evolution. It is, in my opinion a remarkable fact that the Born-Markov master equation can be used to churn out a master equation in the Lindblad form. Similarly we can appreciate Lindblad’s work a lot more when we question the validity of the Born-Markov approximation. The moral of the story is that
Born-Markov and Lindblad complement each other and only by knowing both results do we get a complete picture of quantum Markovian dynamics. I believe therefore quantum
optics textbooks should present a more balanced approach to open systems theory by stating Lindblad’s theorem because this is when knowing a mathematical theorem actually allows one to appreciate the physics more.

*The Lindblad form should also be credited to Gorini, Kossakowski, and Sudarshan but I’ll refer to it as Lindblad just for convenience.

** I use the word “leads” because the master equation in the Born-Markov approximation alone does will not directly give us an equation in the Lindblad form.
In quantum optics one usually need to make a further approximation called the rotating-wave approximation to obtain the Lindblad form.

What does it mean to understand in physics?

Every time we learn a new theory which is supposed to explain some physical phenomena we try to understand what is behind its axioms, why certain definitions were introduced and what is the physical meaning of the derived theorems. Let’s think about what it really means to understand and what we want to achieve from our study. Maybe it is fair enough just to be convinced about the significance of the theory without any deeper understanding.

Investigating a problem usually goes for searching for the logical reasoning between facts we find to be a cause of observed effects and using for this physical theories as a framework. To give an example, if we consider the planet movement we expect to take into account the gravitational interaction. Then we are able to foresee their future position but do we really get the underlying physics? The answer is yes; if we only wanted to describe another planetary system, we would succeed in doing so again. However, we cannot be satisfied, at this point we still do not know the interaction mechanism and cannot justify the formula for gravitational force.

With no doubt we should always examine the reasons for which the theory introduces its concepts. It is the best way to get to know all restrictions of the theory and to take a critical look at all made assumptions. On the other hand we will always face the wall of ideas and axioms, which not necessarily have to be intuitive.  The definition of kinetic energy was introduced by classical mechanics and became a part of the everyday language but still it is a purely theoretical notion. So it looks like we are condemned to play the game with known rules, we may even enjoy the game but will never find the explanation of all principles. We have to accept that there is a limit which we will never cross.

Quite often we just accept the validity of formulas and expressions proposed by other researchers. Maybe the reason of this attitude comes from the academic pressure. Having just a taste of the new theory we might be in the position to quickly identify some problems that could be solved within it. Things are fine as long as we remember to take the next step – going into the details of the theory. However, we can also say that using the elaborated model without considering all technical details does not have to indicate we are lazy. We may just want to get the new results based on what has been done by our predecessors.

It is very subjective when we can say, “OK, I’ve understood.” Maybe it is when we are ready to explain certain problems to others or we are just convinced about our knowledge and thanks to it we can predict the results of experiment. Most likely we will never succeed in our attempts to understand the whole nature. Thus the statement  “Still I don’t understand” takes the advantage as it drives our scientific progress.