From Schrödinger’s cat to quantum computers (II)

From analog computers to quantum computers

In this second part of the talk, we move on to discuss quantum computers. They are fashionable, they’re cute. You can boast in bars that you work in quantum computers and you’ll get free beers. But… are they, really, more powerful than their classical counterparts? I have devised a way to explain what is the main difference, and why they might well be more powerful. And it uses an old and forgotten concept: that of analog computer.

What is an analog computer? It is a machine designed to solve a certain computational problem, using physics. For example, the Antikythera mechanism. It is a device which was found in a sunk Greek ship from the Hellenistic times. It has some gears which, when spinning, represented the motion of planets, and helped predict their positions in the sky. In a certain way, the astrolabe is also an analog computer.

The Antikythera mechanism, an early analogue computer.

But let us give a simpler example. Imagine that you must order a large set of numbers, from lowest to highest. You can design an “ordering computer”, in the following way: get some spaghetti and cut each of them to a length corresponding to one of the numbers. Then, you take the full bunch and hit the table with it, flattening its bottom. Now, you only have to pick the spaghetti in order: first, second, third…

Another cute example is the problem of finding the most distant cities in a roadmap. I give to you a list of cities, and the distances those which are linked by a road. An analog computer can be built with a long thread. We cut it to pieces representing the roads joining
each pair of cities, and them we tie them in a way resembling the full roadmap. The knots, of course, represent the cities. Now we pick the roadmap by one of the knots, and let the rest hang freely. We look at the knot which lies the lowest. Now we pick it, and let the rest hang freely. In a few iterations we converge to a cycle between to knots. Those are the most distant cities in the roadmap.

And another one! Consider a 2D map on which we are given the positions of a few cities. We are asked to design the road network of minimal length which joins them all. Sometimes it will be convenient to create crossroads in the middle of nowhere. This is called the Steiner tree problem, and it is considered a “hard” problem. There is a way to solve it fast with an analog computer: we get a board and put nails or long pins on it representing the cities. Now we immerse the full board into soapy water and take it out very slowly. A soap film will have developed between the nails, with some “crossroads”, joining all pins. If we do it slowly enough, the film will have the minimum energy, which means that the total roadmap will have the minimal length. This idea is important: we have converted our computational problem into one that Nature can solve by “minimizing the energy”.

Experiments by Dutta and coworkers, http://arxiv.org/abs/0806.1340

Our paradigm problem to solve will be the spin glass problem. It sounds like a technical problem, but I have a very nice way to explain it: how to combine your goals in life and be happy. Well, we all want things which simply do not go together easily: work success, health, true love, going out with the buddies, children, etc. Let us represent each of those goals as a small circle, and give a weight to each of them. Now I draw “connection” lines among the goals, which can be positive, if they reinforce each other, or negative, if they are opposite. Each line has a “strength”. So, for example, “having children” is highly incompatible with “going out with the buddies”, but it is strongly compatible with “finding true love”. Which is the subset of those goals I should focus on in order to maximize my happiness?

Each node is a “target”, blue links mean “compatible”, and red means “incompatible”.

I will tell you how to solve this problem, which is truly hard, using an analog computer. We put atoms in each node, and we make “arrow up” mean “focus on this goal”, while “arrow down” will mean “leave it”. Now, experimental physicists, which are very clever guys, they know how to “lay the cables” so that the energy is minimized when the system represents the maximum happiness. Cool.

No, not so cool. The problem is the “false minima”. Imagine that you’re exploring the bottom of the ocean and you reach a very deep trench. How can we know that it is, indeed, the deepest one? Most of the time, the analog computer I just described will get stuck in the first trench it finds. And, believe me, there are many. It’s just the story of my life: I know I can do much better, but… I would have to change so many things, and intermediate situations are terrible. But today I feel brave, I really want to know how to be happy.

Quantum mechanics comes to our help. Remember that atoms are quantum, and that their arrows can point in some other directions, not just up or down. So I put my analog computer, made of real (quantum) atoms inside a giant magnet, which forces all spins to point rightwards. Remember: , so now each one has 50% probabilities of pointing up and pointing down, like Schrödinger’s cat. Maximal uncertainty. I know nothing about what to do. Let us lower slowly the power of the magnet. Nature always wants to minimize the energy, so we pass through some complex intermediate states, highly “entangled”, in which some atoms decide early which position to choose. They are the “easy atoms”, for which no competition with other atoms exist. The “difficult atoms” are the ones in which you have more doubts, and they stay in a “catty” state for longer time. When, finally, the power of the magnet has come to zero, all atoms must have made up their minds. We only have to read the solution, which is the optimal happiness one.

Sure? It all depends on the speed at which we have turned the magnet off. If I am greedy and do it fast, I will ruin the experiment, and reach any “false minimum” of the energy. So, a false maximum of the happiness. All this scheme is called “adiabatic quantum computation”. For physicists, “adiabatic” means “very slow”.

How slow should I go? Well, this is funny. Apparently, most of the time it’s not very important. But there is a critical moment, a certain “phase transition” point, when the entanglement between the atoms is maximal. Then, it is crucial to advance really slowly. As an analogy, think that you have to take a sleeping baby from the living room to his cradle. There is always a fatidic moment, when you have to open the damned doorknob. If you are unlucky, you may have more than one. But, for sure, at least you’ll have one.

And, what if we’re so clever that we have left all doors open? That’s what worries us physicists most. There is a conjecture, that there is some kind of “cosmic censorship”, that will impose a closed door in the path of every difficult problem. Nature might be evil, and has put obstacles to the possibility of solving difficult problems too fast. It would be a new limit: the unsurmountable speed of light, the unstoppable increment of entropy… and now that? It is worth to pay attention: the next years will be full of surprises.

This is the second part of a lecture I originally delivered in the streets of Madrid for the “Uni en la calle”, to protest the budget cuts in education and science in Spain, in March 9, 2013. And, later at a nice high school in Móstoles.

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From Schrödinger’s cat to quantum computers (I)

From Schrödinger’s cat to entangled cats

If you are also fans of The Big Bang Theory, you will be aware of Penny and Sheldon’s discussions about Schrödinger’s cat. Penny wants to know whether she should hook up with Leonard, and Sheldon tells her that, in 1935, Erwin Schrödinger designed a mental experiment in which a cat was put inside a closed box with a vial of poison which can be opened at random times. You may not know whether the cat is alive or dead until you open it. Penny thinks that the lesson is that she should try, and only then she will know. But, really, Sheldon only wanted to get rid of her. The question, as all human interaction, was completely irrelevant to him.

I have a board. If you like boards, this is my board.

Although I love the series, the explanation about Schrödinger’s cat is lame. You put a cat in a box and a vial of poison. There’s 50% chance that the vial opens and the cat dies. According to our intuition, the real state of the system is one of them: alive-cat or dead-cat. Since we don’t know which one it is, we represent our knowledge with probabilities:

But that’s not quantum mechanics! Quantum mechanics is far more weird, and tell us that the cat may be alive and dead at the same time. We represent it this way:

(That notation, , is called a “ket”… yes, we physicists are very fond of funny notations.) If, while in that state, you open the box, the cat is forced to choose. With 50% probabilities, it becomes an alive-cat, and with 50% a dead-cat. But then, how is it different from before!?  Because the “alive-and-dead-cat” is a new “catty state” that we may represent this way:

and which has different properties.

Well, with cats this doesn’t really work. We tried, but they move a lot, and miaow, and scratch. We better try with atoms, which are far more peaceful. Most atoms behave like small magnets, and their magnetization can be thought of as a small arrow, called “spin”, pointing in any direction, something like this:

We have our (cat-like) atom in a closed box, with its little arrow pointing in any direction. With cats, you may ask: “are you alive or dead?”, and it gives you an answer. With atoms you may ask, for example: “is your little-arrow (spin) pointing up or down?” Of course, not only for the vertical direction. You just pick up any direction and ask, but let’s say that the vertical direction is clear enough. So, you may have the atom in state , and the answer will be “up”, or and the answer will be “down”. But what happens if you mix them? You can have the state . Then when you ask “Is your little-arrow pointing up or down?”, the atom chooses with 50% probability and with 50%.

(By the way, if someone is thinking of erotic analogies, flash news: we physicists have already thought of all of them.)

But I told you that is more than just 50% up and 50% down. Let’s change direction. Now, instead of asking about up or down, we ask “is your little-arrow (spin) pointing rightwards or leftwards?” The atom answers “rightwards” with certainty. 100% probability!! So… that was the point!! It answered randomly when asked about up or down, because it was pointing to the right!! Is there any way to prepare the atom so it points always leftwards? Yeah, we write . And the same happens if you ask the other way round: if you have and ask “are you pointing left or right?”, it will answer randomly. Wrong question, random answer.

But let’s come back to cats. We can go beyond Schrödinger and put two killer cats in the same box. They hate each other, and only one will survive. The quantum state can be written as

but… it could be the other way round! It might be

Classically, we would have 50% of each, But, in quantum mechanics, we can have the state

I put both signs because both are possible. It depends on the cat breed, I think.

But, I insist, that’s hard to do with cats. Do it at your own risk. With atoms, it’s a whole different story. We may prepare atoms such that their little-arrows (spins) point, for sure, in opposite directions. Let’s say that they are in the state

This state suffers from what we call entanglement. And very weird things happen to it. That was studied by Einstein and some of his buddies, called Podolsky and Rosen, in 1935 (also) (yeah, good year), when they showed that we could do the following. Take the box containing both atoms and split it in half, making sure that a single atom stays in each half-box. Now, take one of the boxes very far away. When you ask one of the atoms “are you pointing up or down?”, you don’t know what the answer is going to be, because it chooses randomly. If we do it with cats, you don’t know if the box you’ve kept contains the dead or the alive cat. But let’s assume that you get the answer “up”. Then we know what will the other atom reply when asked whether its arrow  points up or down. It will say “down”.

The surprise comes when you ask the atom: “is your arrow pointing left or right?” Its answer will also come randomly, 50% right and 50% left. I am not going to justify that, just believe me. But if you ask the same question to the other atom, no matter how far away it is, its answer will be the opposite!!! You may ask about any direction, and both atoms will give you opposite answers. The question that we may ponder is, of course… how does the second atom know what was measured on the first? Apparently, entangled atoms hold a bond that, like good loves and good hates, survives distance.

This is the first part of a lecture I delivered in the street in Madrid, as a part of the “Uni en la calle” program to protest the budget cuts in education and science in Spain, on March 9, 2013. I have delivered it also at Manuela Malasaña high school. Thanks to you all, guys!

It’s hot when I accelerate!

Unruh effect and Hawking radiation

Let us discuss one of the most intriguing predictions of theoretical physics. Picture yourself moving through empty space with fixed acceleration, carrying along a particle detector. Despite the fact that space is empty, your detector will click sometimes. The number of clicks will increase if you accelerate further, and stop completely if you bring your acceleration to zero. It is called Unruh effect, and was predicted in 1976.

That’s weird, isn’t it? Well, we have not even scratched the surface of weirdness!

So, more weirdness. The particles will be detected at random times, and will have random energies. But, if you plot how many particles you get at each energy, you’ll get a thermal plot. I mean: the same plot that you would get from a thermal bath of particles at a given temperature T. And what is that temperature?

That is called the Unruh temperature. So nice! All those universal constants… and an unexpected link between acceleration and temperature. How deep is this? We will try to uncover that.

In our previous Physics Napkin we discussed the geometry of spacetime felt by an accelerated observer: Rindler geometry. Take a look at that before jumping into this new stuff.

Has this been proved in the laboratory?

No, not at all. In fact, I am working, with my ICFO friends, in a proposal for a quantum simulation. But that’s another story, I will hold it for the next post.

So, if we have not seen it (yet), how sure are we that it is real? How far-fetched is the theory behind it? Is all this quantum gravity?

Good question! No, we don’t have any good theory of quantum gravity (I’m sorry, string theoreticians, it’s true). It’s a very clear conclusion from theories which have been thoroughly checked: quantum field theory and fixed-background general relativity. With fixed background I mean that the curvature of spacetime does not change.

Detecting particles where there were none… where does the energy come from?

From the force which keeps you accelerated! That’s true: whoever is pushing you would feel a certain drag, because some of the energy is being wasted in a creation of particles.

It’s hot when I accelerate!! Ayayay!!!

I see appeared in the formula for the Unruh temperature. Is it a purely quantum phenomenon?

Yes, although there is a wave-like explanation to (most of) it. Whenever you move with respect to a wave source with constant speed, you will see its frequency Doppler-shifted. If you move with acceleration, the frequency will change in time. This change of frequency in time causes makes you lose track of phase, and really observe a mixture of frequencies. If you multiply frequencies by hbar, you get energies, and the result is just a thermal (Bose-Einstein) distribution!

But, really… is it quantum or not?

Yes. What is a particle? What is a vacuum? A vacuum is just the quantum state for matter which has the minimum energy, the ground state. Particles are excitations above it. All observers are equipped with a Hamiltonian, which is just a certain “way to measure energies”. Special relativity implies that all inertial observers must see the same vacuum. If the quantum state has minimal energy for an observer at rest, it will have minimal energy for all of them. But, what happens to non-inertial observers? They are equipped with a Hamiltonian, a way to measure energies, which is full of weird inertial forces and garbage. It’s no big wonder that, when they measure the energy of the vacuum, they find it’s not minimal. And, whenever it’s not minimal, it means that it’s full of particles. Yet… why a thermal distribution?

Is all this related to quantum information?

Short story: yes. As we explained in the previous post, an accelerated observer will always see an horizon appear behind him. Everything behind the horizon is lost to him, can not affect him, he can not affect it. There is a net loss of information about the system. This loss can be described as randomness, which can be read as thermal.

Long story. In quantum mechanics we distinguish two types of quantum states: pure and mixed. A pure quantum state is maximally determined, the uncertainty in its measurements is completely unavoidable. Now imagine a machine that can generate quantum systems at two possible pure states A and B, choosing which one to generate by tossing a coin which is hidden to you. The quantum system is now said to be in a mixed state: it can be in any two pure states, with certain probabilities. The system is correlated with the coin: if you could observe the coin, you would reduce your uncertainty about the quantum state.

The true vacuum, as measured by inertial observers, is a pure state. Although it is devoid of particles, it can not be said to be simple in any sense. Instead, it contains lots of correlations between different points of space. Those correlations, being purely quantum, are called entanglement. But, besides that, they are quite similar to the correlations between the quantum state and the coin.

When the horizon appears to the accelerated observer, some of those correlations are lost forever. Simply, because some points are gone forever. Your vacuum, therefore, will be in a mixed state as long as you do not have access to those points, i.e.: while the acceleration continues.

Where do we physicists use to find mixed states? In systems at a finite temperature. Each possible pure state gets a probability which depends on the quotient between its energy and the temperature. The thermal bath plays the role of a hidden coin. So, after all, it was not so strange that the vacuum, as measured by the accelerated observer, is seen as a thermal state.

Thermal dependence with position

As we explained in the previous post, the acceleration of different points in the reference frame of the (accelerated) observer are different. They increase as you approach the horizon, and become infinite there. That means that it will be hotter near the horizon, infinitely hotter, in fact.

After our explanation regarding the loss of correlations with points behind the horizon, it is not hard to understand why the Unruh effect is stronger near it. Those are the points which are more strongly correlated with the lost points.

But from a thermodynamic point of view, it is very strange to think that different points of space have different temperatures. Shouldn’t they tend to equilibrate?

No. In general relativity, in curved spacetime we learn that a system can be perfectly at thermal equilibrium with different local temperatures. Consider the space surrounding a heavy planet. Let us say that particles near the surface at at a given temperature. Some of them will escape to the outer regions, but they will lose energy in order to do so, so they will reach colder. Thus, in equilibrium systems, the temperature is proportional to the strength of gravity… again, acceleration. Everything seems to come together nicely.

And Hawking radiation?

Hawking predicted that, if you stand at rest near a black hole, you will detect a thermal bath of particles, and it will get hotter as you approach the event horizon. Is that weird or not? To us, not any more. Because in order to remain at rest near a black hole, you need a strong supporting force behind your feet. You feel a strong acceleration, which is… your weight. The way to feel no acceleration is just to fall freely. And, in that case, you would detect no Hawking radiation at all. So, Hawking radiation is just a particular case of Unruh effect.

There is the feeling in the theoretical physics community that the Unruh effect is, somehow, more fundamental than it seems. This relation between thermal effects and acceleration sounds so strange, yet everything falls into its place so easily, from so many different points of view. It’s the basis of the so-called black hole information paradox, which we will discuss some other day. There have been several attempts to take Unruh quite seriously and determine a new physical theory, typically a quantum gravity theory, out of it. The most famous may be the case of Verlinde’s entropic gravity. But that’s enough for today, isn’t it?

For references, see: Crispino et al., “The Unruh effect and its applications”.

I’ll deliver a talk about our proposal for a quantum simulator of the Unruh effect in Madrid, CSIC, C/ Serrano 123, on Monday 14th, at 12:20. You are all very welcome to come and discuss!

Feeling acceleration (Rindler spacetime)

This is the first article of a series on the Unruh effect. The final aim is to discuss a new paper on which I am working with the ICFO guys, about a proposal for a quantum simulator to demonstrate how those things work. We are going to discuss some rather tough stuff: Rindler spacetime, quantum field theory in curved spacetime, Hawking radiation, inversion of statistics… and it gets mixed with all the funny stories of cold atoms in optical lattices. I’ll do my best to focus on the conceptual issues, leaving all the technicalities behind.

Our journey starts with special relativity. Remember Minkowski spacetime diagrams? The horizontal axis is space, the vertical one is time. The next figure depicts a particle undergoing constant acceleration rightwards. As time goes to infinity, the velocity approaches c, which is the diagonal line. But also, as time goes to minus infinity, the velocity approaches -c. We’ve arranged things so that, at time t=0, the particle is at x=1.

Minkowski diagram of an accelerated particle.

Now we are told that the particle is, really, a vehicle carrying our friend Alice inside. Since the real acceleration points rightwards, she feels a leftwards uniform gravity field. Her floor, therefore, is the left wall.

Alice in her left. Acceleration points rightwards, “gravity” points leftwards.

Are you ready for a nice paradox? This one is called Bell’s spaceship paradox. Now, imagine that Bob is also travelling with the same acceleration as Alice, but starting a bit behind her. Their trajectories can be seen in the figure

Alice and Bob travel with the same acceleration. Their distance, from our point of view, is constant.

From our point of view, they travel in parallel, their distance stays constant through time. So, we could have joined them with a rigid bar from the beginning. Wait, something weird happens now. As they gain speed, the rod shrinks for you… This is one of those typical paradoxes from special relativity, which only appear to be so because we don’t take into account that space and time measures depend on the point of view. This paradox is readily solved when we realize that, from Alice’s point of view, Bob lags behind! So, in order to keep up with her, and keep the distance constant, Bob should accelerate faster than her!

So, let us now shift to Alice’s point of view. Objects at a fixed location at her left move with higher acceleration than she does, and objects at her right move with lower acceleration. Her world must be pretty strange. How does physics look to her?

One of the fascinating things about general relativity is how it can be brought smoothly from special relativity when considering accelerating observers. In order to describe gravity, general relativity uses the concept of curved spacetime. In order to describe how Alice feels the world around her we can also use the concept of curved spacetime. It’s only logical, Mr Spock, since the principle of equivalence states that you can not distinguish acceleration from a (local) gravity field.

Fermi and Walker explained how to find the curved spacetime which describes how any accelerated observer feels space around her, no matter how complicated her trajectory is. The case of Alice is specially simple, but will serve as an illustration.

The basic idea is that of tetrad, the set of four vectors which, at each point, define the local reference frame. In German, they call them “vier-bein”, four-legs, which sounds nerdier. Look at the next figure. At any moment, Alice’s trajectory is described by a velocity 4-vector v. Any particle, it its own reference frame, has a velocity 4-vector (1,0,0,0). Therefore, we define Alice’s time-vector as v. What happens with space-vectors? They must be rotated so that the speed of light at her point is preserved. So, if the time-vector rotates a given angle, the space-vector rotates the same vector in the opposite direction, so the bisector stays fixed.

The local frames of reference for Alice, at two different times.

Now, each point can be given a different set of “Alice coordinates”, according to local time and local space from Alice point of view. But this change of coordinates is non-linear, and does funny things. The first problem appears when we realize that the space-like lines cross at a certain point! What can this mean? That it makes no sense to use this system of coordinates beyond that point. That point must be, somehow, special.

In fact, events at the left of the intersection point can not affect Alice in any way! In order to see why, just consider that, from our point of view, a light-ray emmited there will not intersect Alice’s trajectory. Everything at the left of the critical point is lost forever to her. Does this sound familiar? It should be: it is similar to the event horizon of a black hole.

Red: what Alice can’t see. Green: where Alice can’t be seen.

Let us assume that you did all the math in order to find out how does spacetime look to Alice. The result is called Rindler spacetime, described by the so-called Rindler metric. In case you see it around, it looks like this

Don’t worry if you don’t really know what that means. Long story short: when Alice looks at points at her left (remember, gravity points leftwards), she sees a lower speed of light. Is that even possible? That is against the principle of relativity, isn’t it? No! The principle of relativity talks about inertial observers. Alice is not.

So, again: points at her left have lower speeds of light. Therefore, relativistic effects are “more notorious”. Even worse: as you move leftwards, this “local speed of light” decreases more and more… until it reaches zero! Exactly at the “special point”, where Alice coordinates behaved badly. What happens there? It’s an horizon! Where time stood still.

The world for Alice, Rindler spacetime: speed of light depends on position, and becomes zero at the horizon.

Imagine that Alice drops a ball, just opening her hand. It “falls” leftwards with acceleration. OK, OK, it’s really Alice leaving it behind, but we’re describing things from her point of view. Now imagine that Bob is inside the ball, trying to describe his experiences to Alice. Bob just feels normal, from his point of view… he’s just an inertial observer. But Alice sees Bob talking more and more slowly, as he approaches the horizon. Then, he friezes at that point. Less and less photons arrive, and they are highly redshifted (they lose energy), because they had to climb up against the gravitational potential. Finally, he becomes too dim to be recognized, and Alice loses sight of him.

That description would go, exactly, for somebody staying fixed near a black hole dropping a ball inside it. The event horizons are really similar. In both cases, the observer is accelerated: you must feel an acceleration in order to stay fixed near a black hole! As Wheeler used to say, the problem of weight is not a problem of gravitation. Gravitation only explains free fall. The problem of weight is a problem in solid state physics!!

For more information, see Misner, Thorne and Wheeler’s Gravitation, chapter 6. It’s a classic. I wish to thank Alessio, Jarek and Silvia for suffering my process of understanding…

How many dimensions did you say?

My friends and colleagues from ICFO, Alessio Celi and Maciej Lewenstein (along with O. Boada and J.I. Latorre), have just published a surprising article in Physical Review Letters, which appears in its Synopsis. What is the big deal?  They propose a route to simulate the behaviour of quantum matter in extra dimensions. The idea is extremely simple once it has been understood. But let me start by telling you what is the framework, I mean: what do I mean by simulation.

Consider that you would like to design a new material, which you want to have some nice properties, such as superconductivity, or a given response to magnetic fields… whatever. Most of these properties are given by the quantum behaviour of electrons inside the crystal. The problem is that the behaviour of interacting electrons in a given system is very hard to predict theoretically, using either pencil and paper calculations or huge supercomputers. What to do, then?

When aeronautical engineers design a new airplane, they do some complex calculations. After that, since they can not rely completely on them, they make a model plane and test it in a wind tunnel. They perform a controlled simulation. If the solution to their equations
coincides with the results of the simulation, then they feel confident about the airplane, and the manufacture procedure begins.

Quantum simulators follow the same idea as the wind tunnel and the model airplane. Set up many laser beams, making up a 3D lattice. The lattice spacing will be much larger than in crystalline solids, more than one μm. Now, instead of electrons, we put some ultra-cold atoms. But, I hear you say, atoms are not elementary particles, unlike electrons. There is a nice response to that: anything is an elementary particle until you hit them hard enough! In other words: atoms behave totally like elementary particles if the temperature and the interaction energy is low enough. If they have total spin 1/2, then the atoms are fermionic and behave much like electrons.

So, you have all the elements. Now, let us check a possible design for a material with some concrete properties: set up your optical lattice, put some ultra-cold atoms in there and see. The best part is that if you do not strike oil at the first attempt, you can always change your parameters almost on-the-fly and try again: tune the lasers, heavier atoms… whatever.

Now that we know what a quantum simulator is, let us focus on the novel part: the work of my colleagues. Many speculative theories in physics require the existence of extra-dimensions. If they exist, then their extension must be really small not to appear in ordinary experiments. I do not mean that those theories should be taken seriously, only that we might desire to find out what would be the implications!

Imagine that we prepare our optical lattice and leave our atoms inside. Atoms jump from a cell to the next tunneling through the laser beam. Now, consider atoms that can be in N different internal states, which differ, for example, in the nuclear spin direction. So to speak, N atomic flavours which are nearly indistinguishable. Label the internal atomic states from 1 to , and arrange things so that atoms can only move from state to state or . Now, by tuning up the laser intensities, we can make this movement in internal state to be exactly as movement in any other direction!

The image shows in blue the atoms with flavour 1, and in pink those with flavour 2. An atom at site can jump up, right, back… but it can also change flavour. And that jump would correspond to a movement in the fourth dimension. Of course, the extension of this fourth dimension is extremely reduced if we have only two flavours. In general, we will not be able to achieve huge sizes, but this is not a problem since, as we stated, the extra dimension, if it exists, must be extremely small.

For example, we can arrange a single atom in a given cell, with a given internal state, and let it evolve freely. After some time, it will be in another cell and with another internal state. This internal state will mark how much it has moved in the extra-dimension.

V.I. Arnold, one of the great masters, once said that mathematics is the part of physics where experiments are cheap. Well, the cost of the mathematical experiment must always be compared to the cost of the real one. Using an expensive supercomputer to follow the behaviour of all the atoms of a stone as it falls to the ground does not seem to be a huge saving. But using ultracold atoms in an optical lattice to simulate 4D space qualifies much better… most of all because we are not aware of any other experimental setup! :)

More information about quantum simulators can be gained here or here.

Neutrino jokes (and more)

To be honest, I do not expect much from all this neutrino fuss. I bet that, when the dust settles down, c will remain majestic in her velocity throne and forgive magnanimously our misgivings. Why? OK, first, because superluminal neutrinos would produce a vast amount of electron-positron pairs in their way (see this paper by Cohen and Glashow), which has never been observed.  But, more importantly, because, as Alvaro de Rújula once said, “You must bet so that losing becomes the most intersting option”. That’s what xkcd said, using different words:

But, in any case, the best offspin from this story are few nice neutrino jokes that have come to stay among us:

MY ALL-TIME TOP SIX NEUTRINO JOKES

• A neutrino. “Who’s there?” Knock-knock!
• The bar-tender: “We don’t serve tachyons in here”. A neutrino comes into a bar.
• A neutrino and a photon come into a bar. For the next 60 nanoseconds, the neutrino complains about how dark it is.
• What does a neutrino do in an optical fiber? Honk the photons!
• A neutrino boyfriend: interacts weakly, goes through you without you noticing and ends before you even started.
• To reach the other side. Why did the neutrino cross the road?

And, profiting from the physics-jokes-revival, here you have two other physics jokes I didn’t know:

– Researchers from INFN have found traces of the elusive Berluschino, the supersymmetric partner of Berlusconi. As opposed to the original, it’s tall, honest and believes in democracy. Unfortunately, it is extremely short-lived in the current Italian political environment.

– Schrödinger’s cat comes into a bar. And doesn’t.

BONUS. I just invented three out of all those jokes. Can you tell which?

Rough is beautiful (sometimes)

No posts for three weeks… you know, we’ve been revolting in Spain, and there are times in which one has to care for politics. But physics is a jealous lover… :)

So, we have published a paper on kinetic roughening. What does it mean? OK, imagine that, while your mind is roaming through some the intricacies of a physics problem, the corner of your napkin falls into your coffee cup. You see how the liquid climbs up, and the interface which separates the dry and wet parts of the napkin becomes rough. Other examples: surface gowth, biological growth (also tumors), ice growing on your window, a forest fire propagating… Rough interfaces appear in many different contexts.

We have developed a model for those phenomena, and simulated it on a computer. Basically, the interface at any point is a curve. It grows always in the normal direction, and the growth rate is random. The growth, also, is faster in the concavities, and slower in the convex regions. After a while, the interfaces develop fractal morphology. I will show you a couple of videos, one in which the interface starts out flat, and another one in which it starts as a circle. The first looks more like the flames of hell, the second more like a tumor.

The fractal properties of those interfaces are very interesting… but also a bit hard to explain, so I promise to come back to them in a (near) future.

The work has been done with Silvia Santalla and Rodolfo Cuerno, from Universidad Carlos III de Madrid. Silvia has presented it at FisEs’11, in Barcelona, a couple of hours ago, so I got permission at last to upload the videos… ;) The paper is published in JSTAT and the ArXiv (free to read).

Let me count the ways…

Alice was so bored, waiting for a message from Bob, that she started to play with the five white rabbits she had got from the Queen of Hearts. She tried to figure out in how many ways she could split her rabbits in groups, like 5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1, so, for 5 rabbits, 7 ways.

She decided to count the partitions in which no group contained more than 3 rabbits… in our example, there are 5. And then, she counted the partitions with no more than 3 groups. Amazingly, although the groups were not the same, the two numbers coincided, also 5.

Alice wondered… She wonders all the time (why?). Is that a coincidence?

One plus one plus one plus one...

π=80

A few unrelated questions around π…

• Why is it true that  π=80?
• Why on Earth did we define π as we did, instead of giving a nice symbol to 2π? Life would be much easier… So many less factors 2 in our books… A quadrant would be just  π/4, not the nonsensical π/2… Can you see any notational advantage? Read this for more info.
• Do you recognize this sequence: 3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, …?
• Why should I have posted this yesterday?

OK, let’s keep it short. And thanks to S.N. Santalla…

Update (March 17) My birthday date appears at position 45,260,128 of π, not counting the initial 3. When was I born? ;) Hint. (Via Pepe Aranda) Moreover: possession of all digits of π makes you infringe all known copyright laws… Do you know why?

Why is g so close to π squared?

The hard facts: (a) The acceleration of gravity on Earth is g ~ 9.8 m/s²; (b) π² ~ 9.87.

The question: Is that pure chance?

The naive answer: Sure. Just change the units, the similarity is gone. Just change the planet, the similarity is gone.

Yet… a little bit of historical research tells us that it is not pure chance. How come?

Of course, if there is a connection between the two values, it must be historical, not physical. The similarity between the two values is just on Earth, and with our units. But how is the meter defined? The definition has evolved with time (and in the US they still use units related to the lengths of their extremities… ains…). For a long time, it was one ten-millionth of the length of the Earth’s meridian. So the relation to the Earth is ensured in the definition, no doubt.

No magic involved, just history. It was the French National Assembly, during the Revolution, defining the meter. They wanted a universal definition, and they came up with that one. But it was not the first one… Before, there were others.

As far as we know, it was the marvellous mind of John Wilkins the first to conceive the idea of meter. And what was his definition? No wonder, the length of a seconds pendulum. That means: a pendulum whose period is two seconds. Now, for a bit of physics, remember that, within the small angles approximation, the period of a pendulum is

Now, imagine that we were using Wilkins’ meter. Then with a pendulum of length 1 length-units, we would have a period 2 time-units. Just solve for g and… hey! You get… π².

Wilkins’ idea went all the way down to Huygens, and to Talleyrand, who proposed it to the French revolutionaries. Technical difficulties, mostly the fluctuations of length with temperature, made them change the choice, but nonetheless picking up a close value.

Le jour du mètre est arrivé!