Topological Phases

The development of ultra-cold atomic and molecular gases has raised the possibility of studying topological phases in out-of-equilibrium spin systems. Unlike traditional condensed matter systems, one cannot simply “cool” into a desired topological ground state by decreasing the temperature of a surrounding bath. Rather, preparation must proceed coherently, e.g. by exploiting the quantum adiabatic theorem. This necessitates a detailed knowledge of the phase transitions separating topological states from their short-range-entangled neighbors and requires understanding the interplay between topology, lattice symmetries and out-of-equilibrium dynamics. One particular recent focus of group is the quantum simulation of fractional Chern insulators and quantum spin liquids in lattice-trapped polar molecules.

Transmission electron microscopy (TEM) holds the potential to resolve many questions in life science by providing a close view of the molecular machinery of the cell. However, biological matter is mostly transparent to electron beams, making for poor image contrast. Although biological macromolecules are almost invisible in TEM, their structure is imprinted in the transmitted electron beam as small variations in the phase of the electron wave function.

Both electron microscopy and light microscopy face the problem of visualizing weak phase variations. In optical imaging, this has been solved by the introduction of Zernike phase contrast. In this method, a spatially selective phase retarder is inserted in the beam path to induce a 90o phase shift in the weak scattered wave relative to the strong directly transmitted wave. The interference between the phase-shifted scattered wave and the transmitted wave creates amplitude modulations, giving rise to a visible image.

Phase contrast imaging with an electron microscope requires creating an analogous phase retarder for electron waves, which has proven difficult due to rapid deterioration experienced by any retardation element exposed to the high energy electron beam.

We are working on a radical solution to this problem, using a laser field configuration as a transparent, indestructible, controllable electron wave retarder. Realization of a reliable phase contrast TEM is likely to bring about many discoveries in structural microbiology, and could lead to further applications for laser-based coherent control of the electron wave function.

Team members

Osip Schwartz

Jeremy Axelrod

Daniel Tuthill


  1. Continuous 40 GW/cm2 laser intensity in a near-concentric optical cavity. Osip Schwartz, Jeremy J. Axelrod, Philipp Haslinger, Robert M. Glaeser, and Holger Müller. arXiv 1610.04893