# Quantum particle near an event horizon

As the next episode in our series about the Unruh effect (it gets hot when you accelerate), here can watch a video I have prepared which depicts how a quantum particle behaves near an event horizon.

So, what are we watching? The left and right panels show the spin-down and spin-up wavefunctions for a massless Dirac particle (a massless electron), initially at rest in Rindler spacetime. Colors correspond to phase. Because of the principle of equivalence, there are two alternate physical interpretations:

• You are moving with constant acceleration rightwards. At time t=0 you drop a Dirac particle. It seems to move leftwards, just because you leave it behind. When it gets far away, it slows down. This is due to relativistic time-dilation.
• There is a uniform gravitational field pointing leftwards. That’s why the Dirac particle accelerates in that direction. As it falls, it slows down. This is due to gravitational redshift.

Of course, the interference pattern which develops at the center is just quantum mechanics, nothing else. But when the particle reaches the edges of the box (top, bottom and right), new interference patterns appear which are spureous to our problem. That’s just the handicap of a finite-size simulation.

Nice, ein? This was work we developed at ICFO, Barcelona, along with Maciej Lewenstein, Alessio Celi and Jarek Korbicz. I have just showed it as a premiere during the Quantum gases meeting at CSIC in Madrid.

# Qubism

Scientists tend to be very visual people. We love to understand through pictures. About one year ago, we had one of those ideas which remind you why it’s so fun to be a theoretical physicist… Simple and deep. The idea was about how to represent quantum many-body wavefunctions in pictures. Speaking very coarsely, the high complexity of the wavefunction maps into fractality of the final image.

So, more slowly. As you know, bit can take only two values: 0 and 1. A qubit is a quantum bit, which can be in any linear combination of 0 and 1, like Schrödinger’s cat, which we denote by $|0\rangle$ and $|1\rangle$. In other terms: a qubit is represented by two complex numbers: $|\Psi\rangle = \alpha |0\rangle + \beta |1\rangle$. If you have two qubits, the basic states are four: 00, 01, 10 and 11, so we get

$|\Psi\rangle = \alpha_{00} |00\rangle + \alpha_{01} |01\rangle + \alpha_{10}|10\rangle + \alpha_{11}|11\rangle$

If you add one qubit, the number of parameters doubles. For N qubits, you need $2^N$ parameters in order to specify completely the state! The task of representing those values in a picture in a meaningful way seems hopeless… Our idea is to start with a square and divide it in four quadrants. Each quadrant will be filled with a color associated with the corresponding parameter.

What if we get a second pair of qubits? Then we move to “level-2”: we split each quadrant into four parts, again, and label them according to the values of the new qubits. We can go as deeply as we want. The thermodynamical limit $N\to\infty$ corresponds to the continuum limit.

The full description of the algorithm is in this paper from arXiv, and we have launched a webpage to publish the source code to generate the qubistic images. So, the rest of this blog entry will be just a collection of pictures with some random comments…

This is the ground state of the Heisenberg hamiltonian for $N=12$ qubits. It is an antiferromagnetic system, which favours neighbouring qubits to be opposite (0-1 or 1-0). The main diagonal structures are linked to what we call a spin liquid.

These four pics correspond to the so-called half-filling Dicke states: systems in which half the qubits are 0 and the other half 1… but you do not know which are which! The four pics show the sequence as you increase the number of qubits: 8, 10, 12 and 14.

This one is the AKLT state for N=10 qu-trits (each can be in three states: -1, 0 or 1). It has some nice hidden order, known as the Haldane phase. The order shows itself quite nicely in its self-similarity.

This one is the Ising model in a transverse field undergoing a quantum phase transition… but the careful reader must have realized that it is not fitting in a square any more! Indeed, it is plotted using a different technique, mapping into triangles. Cute, ein?

But I have not mentioned its most amazing properties. The mysterious quantum entanglement can be visualized from the figures. This property of quantum systems is a strong form of correlation, much stronger than any classical system might achieve.

So, if you want to learn more, browse the paper or visit this webpage, although it is still under construction…

With warm acknowledgments to my coauthors: Piotr Midgał, Maciej Lewenstein (ICFO), Miguel I. Berganza and Germán Sierra (IFT), and also to Silvia N. Santalla and Daniel Peralta.

# Quantum dreams

Quantum mechanics, the dreams stuff is made of… (David Moser)

A quantum particle, prisoner in a square box of infinite walls, starts out with minimal energy, which grows and grows, slowly… although, no matter how much energy it gathers, no matter it grows quadratically… it will never escape…

You can also see it as the vibrational modes of a square drum. It looks continuous because I interpolated between them for a smoother visualization…

# Sqpirals

This time the challenge is directly… to find out what’s that! I promise, it’s an extremely simple algorithm. I got the idea in a coffee-talk conversation, as good physics napkins should :), I don’t know if somebody has given them a name. For me, they’re sqpirals…

Njoy…

BTW, the final frame is, I know, my desktop… But I was too lazy to repeat the video! XD

# Drawing knots

A problem came to my desk from the hands of Dani (a nice source of problems, btw): drawing knots on a computer! I messed around the web and found a couple of pstricks… but they were not general enough, so I made up my mind to try my own generator. This way xknots was born.

Xknots reads a file in a certain format and renders a postscript file for a 2D view of the knot. What is a 2D view of the knot? OK, here you have one, which goes by the name of trefoil knot:

First of all, mathematical knots are closed curves on $R^3$. So, no dangling ends. A 2D view is just flattening the knot, and marking at each crossing which thread is above which. My idea was to invent a description rule for each knot. First, we state the number of crossings. So, 3 in our case. In order to understand a crossing, let us look at the next picture:

There are two types of crossings: L and R. All of them have four legs, numbered 1 to 4. So, in order to describe the knot, we give the coordinates of the crossings, along with their type. For example:

(200,100) L

That’s a nice crossing description. We can also, if needed, specify the angles:

(200,100) L A 60 90

The “A 60 90” means that the first leg will point at 60 degrees (counterclockwise) from the X axis, and the angle between the 1 and 2 legs is 90 degrees. Once the crossings are done, we join them with lines, for example:

1-2 3-4

This means: join the 2nd leg of the 1st crossing to the 4th leg of the 3rd crossing. If needed, we can specify how “extended” that line should be. This way,

1-2 3-4 F 2

means that this line wants to go very far away from the shortest path (the default is F 1). The full code for the trefoil is:

3
(120,120) L A 60 120
(180,120) L A 0 120
(150,172) R A 60 60
1-1 3-3
1-2 3-2 F 2
1-3 2-4 F 2
1-4 2-3
2-1 3-1 F 2
2-2 3-4

(all calculated to make a nice equilateral triangle). Here you have a couple of knots more:

The first is the “borromean rings”, the second goes by the funny name of $7_4$… So, you can find the source code (C++ for linux, but pretty standard), more ideas and more explanations at the main webpage of the project…

And thanks to Dani & Alberto!