Interacting Fibonacci anyons in a Rydberg gas

The physics of interacting Fibonacci anyons can be studied with strongly interacting Rydberg atoms in a lattice, when due to the dipole blockade the simultaneous laser excitation of adjacent atoms is forbidden. The Hilbert space maps then directly on the fusion space of Fibonacci anyons. Interactions between anyons are generated and controlled by the intensity and frequency of the excitation laser. Fusion outcomes of neighboring anyons can be determined experimentally via the measurement of three-point correlations among three consecutive atoms. Our work shows that a Rydberg lattice gas constitutes a natural physical platform for the experimental exploration of topological quantum liquids of non-Abelian anyons.

Figure 1

(a) Fusion space. The Hilbert space of a system of $N$ Fibonacci anyons of type $\tau$ is spanned by all possible fusion paths. An example of such a path is given by the symbols contained in the boxes. A fusion path translates into configurations of atoms which are either in the electronic ground state or excited to a Rydberg state. The Rydberg state (ground state) is identified with the fusion outcome 1 ($\tau$). (b) Level scheme and interaction energy. The excitation of a Rydberg atom on site $k$ shifts the energy of Rydberg states of atoms in the neighborhood. The strong interaction between neighboring atoms (here $k$ and $k+1$) excludes their simultaneous excitation to Rydberg states and effectively constrains the Hilbert space of the atomic system to the fusion space of Fibonacci $\tau$ anyons.

Figure 2

(a) Changing the order of the fusion path. The $F$ matrix performs a basis change in fusion space such that the fusion outcome $u$ of the two neighboring $\tau$ anyons becomes explicit. (b) Determination of the fusion outcome via a measurement of the three-point correlation function $\langle {\Pi }_k\rangle$. First one maps the Rydberg state $|r\rangle$ to a stable hyperfine state $|s\rangle$ by a strong laser pulse (${\Omega }_{rs}\gg \left|V_2\right|$) to switch off interactions. The state of the atoms on sites $k\pm 1$ is measured via fluorescence from a closed transition from $|g\rangle$ to a short-lived state $|a\rangle$ (decay rate $\Gamma$). In the absence (presence) of fluorescence, an atom is in state $|s\rangle$ ($|g\rangle$). (c) List of measurement results and the corresponding fusion outcome. From Eq. (4) it follows that if the atoms $k-1$ and $k+1$ are in different internal states the fusion outcome is $\tau$. If the atoms are in the state ${|s\rangle }_{k-1}{|s\rangle }_{k+1}$ one finds $\langle {\Pi }_k\rangle =1$, i.e., fusion to a trivial anyon. If the atoms are in ${|g\rangle }_{k-1}{|g\rangle }_{k+1}$ one has to measure the expectation value of the operator that corresponds to the lower right $2\times 2$ block of ${\Pi }_k$ given in Eq. (4). This block can be written as a rotation of the number operator $n=(1+{\sigma }^z)/2$: $R_{\mathbf{n}}^{†}n{R}_{\mathbf{n}}$. Here $R_{\mathbf{n}}=e^{i\pi \mathbf{n}·\sigma /2}$ with $\sigma =({\sigma }^x,{\sigma }^y,{\sigma }^z)$ and $\mathbf{n}=({\varphi }^{-1/2},0,{\varphi }^{-1})$. Hence, after a rotation of the $k$th atom on the Bloch sphere and a subsequent projective measurement, we find the fusion outcome 1($\tau$) if the atom is in the state $|s\rangle$ ($|g\rangle$).

Figure 3

Low-energy excitation spectra. Energy spectra for $H_0$ (left column) and $H_{\mathrm{eff}}$ (right column) with positive and negative $J$ [referred to as case (i) and case (ii) in the text]. The spectra have been shifted such that the lowest-energy eigenvalues are zero. The dashed lines sketch low-energy dispersion relations and serve as a guide to the eye. The different symbols (green circles, red squares, blue diamonds) indicate different system sizes $N$: top panels (19,21,23); bottom panels (16,19,22).