Coherent generation of photonic fractional quantum Hall states in a cavity and the search for anyonic quasiparticles

We present and analyze a protocol in which polaritons in a noncoplanar optical cavity form fractional quantum Hall states. We model the formation of these states and present techniques for subsequently creating anyons and measuring their fractional exchange statistics. In this protocol, we use a rapid adiabatic passage scheme to sequentially add polaritons to the system, such that the system is coherently driven from $n$- to ($n+1$)-particle Laughlin states. Quasiholes are created by slowly moving local pinning potentials in from outside the cloud. They are braided by dragging the pinning centers around one another, and the resulting phases are measured interferometrically. The most technically challenging issue with implementing our procedure is that maintaining adiabaticity and coherence requires that the two-particle interaction energy $V_0$ be sufficiently large compared to the single-polariton decay rate $\gamma , V_0/\gamma \gg 10N^{2}lnN$, where $N$ is the number of particles in the target state. While this condition is very demanding for present-day experiments where $V_0/\gamma \sim 50$, our protocol presents a significant advance over the existing protocols in the literature.

Figure 1

Schematic of the proposed cavity experiment. The noncoplanar arrangement of the mirrors gives rise to an effective magnetic field for the transverse dynamics of the cavity photons (red beam). These photons are coupled to atoms (white dots) loaded into a transverse plane (transparent disk) of the cavity. As shown in the energy level diagram, the photons couple the ground state $|g\rangle$ and an excited state $|e\rangle$ of the atoms with a collective Rabi frequency $G$. The excited state has a single-photon detuning ${\mathrm{\Delta }}_e$ and lifetime $1/\mathrm{\Gamma }$ and is coupled to a metastable Rydberg level $|r\rangle$ by a control laser (blue beam) with Rabi frequency ${\mathrm{\Omega }}_c$. When the two-photon detuning ${\delta }_2$ is smaller than a linewidth, long-lived Rydberg polaritons form which inherit the transverse photon dynamics and interact strongly with one another. In Sec. 3 we describe how one can drive these polaritons to form $\nu =1/2$ Laughlin states in the transverse plane. An additional laser (green beam) can be used to produce a localized potential for the polaritons via the ac Stark shift. As we show in Secs. 4 and 5, by moving such potentials relative to the polariton cloud, one can create and braid anyonic quasihole excitations in a Laughlin state.

Figure 2

(a) Spectrum of three polaritons in a twisted optical cavity, described by the Hamiltonian in Eq. (4), which illustrates the general features of the $N$-body spectrum. The ground state, highlighted by the circled blue dot, has total angular momentum $L=N(N-1)=6$ and corresponds to the $\nu =1/2$ Laughlin state $|{\mathrm{\Phi }}_N\rangle$. At fixed $L$, the excitation gap from this state is ${\mathrm{\Delta }}_N$, which arises from polariton-polariton interactions. The lowest-energy excitation with higher angular momentum has energy $\varepsilon$, whereas that with lower angular momentum has energy ${\mathrm{\Delta }}_N-N\varepsilon$, where $\varepsilon$ is related to the harmonic confinement and the effective magnetic field. The square- and diamond-shaped dots represent the excited states $|{\mathrm{\Phi }}_N^e\rangle$ and $|{\mathrm{\Phi }}_N^g\rangle$ defined in Sec. 3. (b) Number dependence of the excitation gap ${\mathrm{\Delta }}_N$. For $N\gtrsim 5$, it saturates at $0.6V_0$, where $V_0$ is the interaction energy of two particles in the lowest Landau level with zero relative angular momentum (zeroth Haldane pseudopotential).

Figure 3

Spectrum of states coupled during our driving protocol. The solid blue arrow shows the desired transition from the $n$-particle Laughlin state $|{\mathrm{\Phi }}_n\rangle$ to the ($n+1$)-particle Laughlin state $|{\mathrm{\Phi }}_{n+1}\rangle$ with a resonant frequency ${\omega }_n$. Dashed red arrows show possible undesired transitions to the low-lying excited states $|{\mathrm{\Phi }}_{n+1}^e\rangle$ and $|{\mathrm{\Phi }}_{n+2}^g\rangle$. These transitions are off resonant by the many-body gaps ${\mathrm{\Delta }}_{n+1}$ and ${\mathrm{\Delta }}_{n+2}$.

Figure 4

Matrix elements of the drive between the coupled many-body states. The operator ${\widehat{a}}_{2n}$ annihilates a particle in the angular momentum mode $m=2n$.

Figure 5

(a) Overlap of the system wave function $\left|\mathrm{\Psi }\right(t)\rangle$ with each $N$-particle Laughlin state $|{\mathrm{\Phi }}_N\rangle$ as a function of time during our driving protocol, described by Eqs. (6, 7, 8). Each subsequent plateau corresponds to increasing $N$ by 1. The duration of each sweep is $4\tau$. As before, $V_0$ is the two-particle interaction energy. (b) Cumulative error in the final state preparation as a function of $\tau$ for three different particle numbers, with $c_{\mathrm{\Omega }}=4, c_{\delta }=5.33$.

Figure 6

Polariton density $\rho$ in the Laughlin state $|{\mathrm{\Phi }}_{N=3}\rangle$ (left) and a two-quasihole state $|{\mathrm{\Phi }}_{N=3}^{\text{oo}}\rangle$ (right).

Figure 7

(a) Fidelity of the two-quasihole state preparation with a strong impurity potential ($U_0=20V_0$) as a function of the sweep rate $|{\partial }_t(r_0/l)|$ for different trap frequencies parametrized by $\varepsilon$. Here, $r_0$ is the radial distance of each pinning potential from the center and $l$ is the scaled magnetic length. Dashed vertical lines show where $|{\partial }_t(r_0/l)|=\varepsilon$ for each curve. The final overlap approaches unity for $|{\partial }_t(r_0/l)|\lesssim \varepsilon$ provided $\varepsilon \lesssim {\varepsilon }_{\text{th}}\equiv {\mathrm{\Delta }}_N/\left(2N\right)=V_0/8$ (for $N=3$), where ${\mathrm{\Delta }}_N$ is the many-body interaction splitting shown in Fig. 2. To avoid visual distraction, the curve for $\varepsilon /{\varepsilon }_{\text{th}}=4$ is only shown for $|{\partial }_t(r_0/l)|>0.1V_0$. (b) Fidelity of the two-quasihole state preparation as a function of the strength $U_0$ of the applied potential. Here, $|{\partial }_t(r_0/l)|=\varepsilon /2={\varepsilon }_{\text{th}}/2$ (adiabatic sweep).

Figure 8

Contour plot showing the braiding error $\eta$ when two strong impurity potentials ($U_0=100V_0$), each binding a quasihole at $\pm r_0$, are rotated by $\pi$ at a constant angular speed ${\omega }_b$. Here, $N=3$ and $\varepsilon =0$ (no trap). The vertical band centered around $r_0/l\approx 2.5$ corresponds to edge excitations. Other peaks correspond to bulk resonances. As before, $l$ denotes the scaled magnetic length.

Figure 9

Braiding error $\eta$ vs rotation rate ${\omega }_b$ at different trap frequencies for $N=3, U_0/V_0=100$, and $r_0/l=1.5$. Note that for adiabatic quasihole generation, one must have $\varepsilon \lesssim {\varepsilon }_{\text{th}}\equiv {\mathrm{\Delta }}_N/\left(2N\right)=0.125V_0$ [Figs. 7 and 2].

Figure 10

(a), (b) Polariton density in the $x-y$ plane for the two experiments needed to extract the statistical phase ${\varphi }_s$ associated with exchanging two quasiholes. As in Fig. 6, brighter colors represent higher density and the dark disks correspond to quasiholes bound to potentials. Here $N=4$. In (a) two quasiholes are exchanged while in (b) one quasihole is moved in a circle. (c) Statistical phase inferred from subtracting the geometric phases that would be found in these two experiments. In the thermodynamic limit, with well-separated quasiholes, one expects ${\varphi }_s=\pi /2$. Note that $r_0$ is the radial distance of each quasihole from the center and $l$ is the scaled magnetic length.