Ganesh Swami

The guest speaker for this week was Dr. Dirk Bouwmeester, a scientist from the University of California, Santa Barbara. Most of his expertise is in experimental quantum optics. His website also mentions that he was “involved in the first experimental demonstrations of quantum teleportation, quantum cloning, 3-particle entanglement and stimulated emission of entangled photons.” His personal opinion of these experiments is that they don’t excite him so much – it’s mostly observing what you’d predict theoretically. He was looking for something more challenging.

Disclaimer: I didn’t understand a major portion of his talk. I’m only writing about things that I was able to understand. (If that isn’t obvious already.)

The first topic that made the audience uncomfortable was Penrose’s theories of state reduction in quantum mechanics. People familiar with quantum mechanics know that “objects” on the microscopic level exist in multiple quantum states simultaneously. These states somehow magically “collapse” when an observation is made. Penrose’s explanation for this collapse is due to gravity. When the gravitational force between these super-positioned states become significant, they collapse into one that can be physically measured. Bouwmeester’s team is conducting experiments to test this theory and their basis is that our measurement time resolution far outweighs the energy difference between the two superpositions. Mathematically, this is the Heisenberg uncertainty principle.

The second topic that was new to me was about entanglement of two or more photons. Young’s double slit experiment definitely showed that a single photon can interfere with itself. Most introductory textbooks on optics only treat interference with coherent sources (sources with no phase difference.) In fact, Paul Dirac in his book “The principles of Quantum Mechanics” says

Each photon interferes only with itself. Interference between two different photons never occurs.

Dr. Bouwmeester wrote a Nature article about research that has shown entanglement between two or more photons. You can read more about this at the ScienceWeek archive.

That is all.

Had a chance to hike at the Dennett-Munroe Lake trail last week. It was described to me as “moderately-hard,” but it was one of the hardest I’ve done. All of the 1300m evelation is in the first two hours, after which it levels off. Our exact coordinates were here.

The areas surrounding the lakes had mostly dried up. It has slippery and I fell into a swamp twice. You can see my dirty pants in the one of the photographs.

Thanks to Calin for the pictures.

If we want to compute the fourier series expansion of , then our coefficients are readily computed on paper as:

and

Now we are going to call the functions fourier, foursimp and fourexpand, which obviously calculates the coefficients, simplifies it and expands them into a series. Alternatively, you could call totalfourier which is just the composition of the three functions.

foursimp(fourier(x^2,x,%pi));

This will spit out the coefficients , and . We can now use fourexpand to expand this into a finite series (four terms here).

fourexpand(%o7,x,%pi,4);

Now plot the series approximation along with the original to compare:

plot ([x^2,%o8],[x,-3,3]);

That is all.