Using concepts of geometric orthogonality and linear independence, we logically deduce the form of the Pauli spin matrices and the relationships between the three spatially orthogonal basis sets of the spin-1/2 system. Rather than a mathematically rigorous derivation, the relationships are found by forcing expectation values of the different basis states to have the properties we expect of a classical, geometric coordinate system. The process highlights the correspondence of quantum angular momentum with classical notions of geometric orthogonality, even for the inherently non-classical spin-1/2 system. In the process, differences in and connections between geometrical space and Hilbert space are illustrated.

Diode lasers have many useful properties and have found a variety of uses including CD and DVD players, barcode scanners, laser surgery, water purification, quantum-key cryptography, spectroscopic sensing, etc. Nevertheless, their intrinsic linewidth or the precision of their emitted wavelengths, is not good enough for many cutting-edge applications such as atomic interferometry or high-performance atomic clocks. Using active feedback control, we can narrow the linewidth of a diode laser by not allowing the frequency of emitted light to drift away from a reference value. Nevertheless, such feedback designs are challenging because of a lack of first principles models and difficult sensor dynamics. This brief describes our diode laser system and reports our results identifying the system using black-box techniques, validating the empirical models, and designing controllers to achieve desired performance while preserving stability and satisfying implementation constraints.

We present a laser wavelength meter based on a commercial color sensor chip. The chip consists of an array of photodiodes with different absorptive color filters. By comparing the relative amplitudes of light on the photodiodes, the wavelength of light can be determined. In addition to absorption in the filters, etalon effects add additional spectral features which improve the precision of the device. Comparing the measurements from the device to a commercial wavelength meter and to an atomic reference, we found that the device has picometer-level precision and picometer-scale drift over a period longer than a month.

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