Abstract
We study the quantum dynamics of massive impurities embedded in a strongly interacting two-dimensional atomic gas driven into the fractional quantum Hall (FQH) regime under the effect of a synthetic magnetic field. For suitable values of the atom-impurity interaction strength, each impurity can capture one or more quasi-hole excitations of the FQH liquid, forming a bound molecular state with novel physical properties. An effective Hamiltonian for such anyonic molecules is derived within the Born-Oppenheimer approximation, which provides renormalized values for their effective mass, charge and statistics by combining the finite mass of the impurity with the fractional charge and statistics of the quasi-holes. The renormalized mass and charge of a single molecule can be extracted from the cyclotron orbit that it describes as a free particle in a magnetic field. The anyonic statistics introduces a statistical phase between the direct and exchange scattering channels of a pair of indistinguishable colliding molecules, and can be measured from the angular position of the interference fringes in the differential scattering cross section. Implementations of such schemes beyond cold atomic gases are highlighted, in particular in photonic systems.
Highlights
The discovery of the fractional quantum Hall (FQH) effect in two-dimensional (2D) electron gases under a strong transverse magnetic field [1,2,3] is a cornerstone of modern physics
Capitalizing on previous works, we provide a rigorous derivation of the effective Hamiltonian starting from a controlled Born-Oppenheimer (BO) approximation [55,56], where the positions of the impurities play the role of the slow degrees of freedom, and the surrounding FQH fluid provides the fast ones
The same kind of impurity and a bound quasihole excitation, and makes them collide—e.g., by pushing them against each other via a suitable external potential—the angular dependence of the differential scattering cross section will show a pattern of maxima and minima whose angular position can be directly related to the fractional statistics of the molecules, as we show in Sec
Summary
The discovery of the fractional quantum Hall (FQH) effect in two-dimensional (2D) electron gases under a strong transverse magnetic field [1,2,3] is a cornerstone of modern physics. We illustrate the consequences of this long-range topological interaction in the simplest scattering process where two such objects are made to collide For both hard-disk and dipolar interaction potentials, we calculate the differential scattering cross section for indistinguishable impurities, finding that for large relative momenta, it features alternate maxima and minima due to the interference of direct and exchange scattering channels: Analogously to textbook two-slit experiments, the interference pattern rigidly shifts when the statistical phase that the anyonic molecules acquire upon exchange is varied.
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