# Dawn of a New Semester

posted on 2011-01-13 06:48:47

It's now officially Thursday, which means there are only 3 more days before the new semester starts.  If all goes according to plan, I'll be taking Intro to Quantum, a class on teaching math, and a German theatre/plays course.  My fourth course slot will be filled by independent research, because I got asked if I would like to try for honors in physics, and I certainly would!

My research last semester--the required senior research course--was mostly my chance to learn some techniques and theory of quantum optics and computing.  I spent most of my time learning how to measure quantum states, a process called quantum state tomography, how to construct a quantum-optical apparatus, and a few examples of trying to minimize decoherence--when a wave packet gets connected to its environment.   These were all interesting things to learn, but I did very little to further my project's goal: creating arbitrary spatial mode and polarization states in a pair of entangle photons.

My talk at the end of the semester was titled "Generating Qubits for Quantum Computation with Biphotons", which is the end goal, but I mostly talked about the process of quantum tomography.  This semester, I'm hoping to work on much more.

Last semester, I was working only with polarization, that is, the orientation of the oscillations in the electric and magnetic fields that light is composed of.  This semester, if I can recreate my results from last semester sufficiently well in a new setup, I'll be adding spatial modes into the mix, which will increase the Hilbert space of the system to be 16-dimensional, if I understand the math right...

$$\mbox{Two particles, two polarization states:} \mathcal{H} = \left(\begin{array}{l}|H_1\rangle\\|V_1\rangle\\\end{array}\right) \otimes \left(\begin{array}{l}|H_2\rangle\\|V_2\rangle\\\end{array}\right) = \left(\begin{array}{l}|H_1H_2\rangle\\|H_1V_2\rangle\\|V_1H_2\rangle\\|V_1V_2\rangle\\\end{array}\right)$$

$$\mbox{Two particles, four states:} \mathcal{H}_{pol} \otimes \mathcal{H}_{mode} = \mathcal{H}^{16}$$

Which will require a much larger tomography process to read.  When there were only the two polarization states, a QST required 16 measurements of correlation between the system state and a selected state, and then some fancy linear algebra (not really that fancy, but...) before giving up the density matrix.  That was with a Hilbert space of dimension 4!  With one of dimension 16, I'm afraid it might take a whole 256 measurements... and I'm not sure I'm capable of that.