Fractional Angular Momentum in Cold-Atom Systems

The quantum statistics of bosons or fermions are manifest through the even or odd relative angular momentum of a pair. We show theoretically that, under certain conditions, a pair of certain test particles immersed in a fractional quantum Hall state possesses, effectively, a fractional relative angular momentum, which can be interpreted in terms of fractional braid statistics. We propose that the fractionalization of the angular momentum can be detected directly through the measurement of the pair correlation function in rotating ultracold atomic systems in the fractional quantum Hall regime. Such a measurement will also provide direct evidence for the effective magnetic field resulting from Berry phases arising from attached vortices, and of excitations with a fractional particle number, analogous to the fractional charge of the electron fractional quantum Hall effect.

Figure 1

The pair correlation function $g(r,0)$ of the test particles with one particle fixed at the origin. (a) The density profile for the $z$ particle without any background $w$ particles. (b) The density profile for the $z$ particle in the background of a $\nu =1/2$ FQHE state. (c) Solid lines show the exact pair correlation function at $L=N^2+N+M$ evaluated for the the Hamiltonian in Eq. (7) assuming a high contact interaction and the condition ${\omega }_w>{\omega }_z$; dots show the pair correlation function obtained from the wave function in Eq. (2). The pair correlation function is quoted in units of ${\rho }_0=(2\pi {\ell }^2{)}^{-1}$, where $\ell$ is the magnetic length. The number of $w$ particles, $N$, is as shown.

Figure 2

The peak position as a function of the relative angular momentum $M$ for $\nu =1/2$, $2/3$, and $3/4$ for several values of $N$, with [$N_0$, $N_1$, $N_2$] representing the occupations of the lowest three CF LLs. The plot for the largest $N$ is unshifted; for ease of depiction, for each successive $N$, the plot has been shifted down by 10 units. The black and red dashed lines display the behavior in Eqs. (3) and (6), respectively. The fractional part of the angular momentum $M_{\text{eff}}$ is given by the $x$ intercept of the black dashed line. The most probable separation of pairs of test atoms immersed in a fractional Hall sea of majority atoms thus demonstrates fractionalization of their relative angular momentum.

Figure 3

(a) The change in the peak position $R$ caused by shifting the test particle $z^{\prime }$ off center. The colors correspond to the legend in Fig. 1. The change is negligible for a shift of up to $0.3\ell$. Panels (b) and (c) show peak positions as a function of $M$ and shift size; for clarity, the numbers for each value of the shift have been shifted down by 10 units.