Time-of-Flight Measurements as a Possible Method to Observe Anyonic Statistics

We propose a standard time-of-flight experiment as a method for observing the anyonic statistics of quasiholes in a fractional quantum Hall state of ultracold atoms. The quasihole states can be stably prepared by pinning the quasiholes with localized potentials and a measurement of the mean square radius of the freely expanding cloud, which is related to the average total angular momentum of the initial state, offers direct signatures of the statistical phase. Our proposed method is validated by Monte Carlo calculations for $\nu =1/2$ and $1/3$ fractional quantum Hall liquids containing a realistic number of particles. Extensions to quantum Hall liquids of light and to non-Abelian anyons are briefly discussed.

Figure 1

(a) Quasihole braiding phase ${\varphi }_{\mathrm{br}}$ as a function of the distance $R=|{\mathbf{R}}_1-{\mathbf{R}}_2|$ between two quasiholes for systems of $N=20$ particles at filling $\nu =1/2$ (blue diamonds) and $\nu =1/3$ (red circles), where one of the quasiholes is fixed in the origin ($|{\mathbf{R}}_2|=0$). Error bars represent statistical uncertainties on the data. Density profiles characterizing some two-quasihole states are given for $\nu =1/2$ in (b)–(d) and for $\nu =1/3$ in (e)–(g). The position of the outer quasihole is fixed along the $x$ axis at $x_1/\sqrt{2}{l}_B=1$, 3, 5 in (b)–(d) and at $x_1/\sqrt{2}{l}_B=1$, 4, 6 in (e)–(g).

Figure 2

(a) Statistical phase ${\varphi }_{\mathrm{st}}$ as a function of the distance $R=|{\mathbf{R}}_1-{\mathbf{R}}_2|$ between two quasiholes for systems of $N=20$ particles at filling $\nu =1/2$ (blue diamonds) and $\nu =1/3$ (red circles), where the quasiholes are located at diametrically opposite positions ($x_1=-x_2=R/2$). Error bars are smaller than the symbol size. (b) Density profile for $\nu =1/2$ with quasiholes located along the $x$ axis at $x_1=-x_2=2.5\sqrt{2}{l}_B$. (c) Density profile for $\nu =1/3$ with quasiholes located along the $x$ axis at $x_1=-x_2=3\sqrt{2}{l}_B$.