Quantum Hall effect at with light John Cerne, SUNY at Buffalo, DMR In metals, magnetic fields deflect moving charges to produce an electric field (E H ) perpendicular to the flow of the charges. This phenomenon is known as the Hall effect (QHE) and is critical to characterizing materials as well as fundamental science. Under special conditions, quantized steps in E H as a function of carrier density appear in the DC (zero frequency) Hall effect. Although theoretical predictions suggest that the QHE may persist at higher frequencies, it is surprising that one may be able to still observe this behavior when charges oscillate times per second (1 THz). Our techniques have been highly successful in studying novel materials such as graphene and topological insulators, however our infrared Hall measurements also have revealed novel behavior in conventional materials. In collaboration with B.D. McCombe at UB, the PI initiated and led Faraday rotation measurements on a two-dimensional electron gas formed in a GaAs/AlGaAs heterojunction (one of the first topological insulators when in the QHE regime). We see clear evidence of plateaus in the Faraday rotation in the 2-3 THz (8-12 meV) range near 7T magnetic fields resulting from the integer QHE (see figure on right). This startling effect was predicted by Ref. [1], but surprisingly was never observed in spite of decades of cyclotron resonance measurements on GaAs/AlGaAs heterojunctions. Of particular note is that the measurement is in the optical regime, where photons are being resonantly absorbed and one would not expect to see evidence of the DC/topological QHE. This work is currently under review at Physical Review Letters. The new understanding gained in our infrared Hall measurements on graphene led to new ideas for modulating the polarization of infrared light. In April 2013, we submitted a patent disclosure on an “Infrared to GHz Tunable Optical Modulator.” [1] Morimoto, T., Y. Hatsugai and H. Aoki, Optical Hall Conductivity in Ordinary and Graphene Quantum Hall Systems. Physical Review Letters 103, / /4 (2009). DC QHE measurements (blue line) compared with THz Faraday rotation measurements (red squares at 3.14 THz and purple line at 2.52) vs. filling factor ν (carrier density). The THz data show weak ν=4,8 and 10 as well as strong ν=6 and ν=12 plateau features.

Heisenberg’s uncertainty principle and lockin amplifiers made easy John Cerne, SUNY at Buffalo, DMR Since its discovery almost 100 years ago, the Heisenberg uncertainty principle has baffled and confused many students when they first try to learn it. By looking at how the strides of two people walking side-by- side align after multiple steps, the PI has made a simple analogy that greatly clarifies the uncertainty principle. If the two people have nearly equal stride length, the small difference in strides will only become apparent after they have walked side-by-side for many steps, which is the essence of the uncertainty principle. A narrated video explains this analogy and connects it to the Heisenberg uncertainty principle. The PI has made a similar video to graphically illustrate how a lockin amplifier works. This instrument is critical to sensitive measurements in many fields and is based on the fundamentally important concept of Fourier analysis. These videos can be found at the PI’s webpage and show that lockin detection, Fourier analysis, and Heisenberg's Uncertainty Principle are connected to each other very closely. The PI has been very fortunate to have worked with outstanding undergraduate and graduate students. Mentoring these students is critically important to our research efforts and has been highly rewarding. Chase Ellis, one the PI’s PhD students, won a highly-selective National Research Council Post-Doctoral Fellowship in 2013 for his proposal based on his work on graphene in the PI’s lab. a) b) Videos explaining the Heisenberg uncertainty principle a) and lockin detection b).