Stirring trapped atoms into fractional quantum Hall puddles

We theoretically explore the generation of few-body analogs of fractional quantum Hall states. We consider an array of identical few-atom clusters $(n=2,3,4)$, each cluster trapped at the node of an optical lattice. By temporally varying the amplitude and phase of the trapping lasers, one can introduce a rotating deformation at each site. We analyze protocols for coherently transferring ground-state clusters into highly correlated states, producing theoretical fidelities (probability of reaching the target state) in excess of 99%.

Figure 1

(Color online) Transferring small clusters from nonrotating ground state to $\nu =1∕2$ Laughlin state using rotating quadrupolar $(m=2)$ deformations. Left: interaction energy (in units of $U∕2$) of quantum states of harmonically trapped two-dimensional clusters as a function of total angular momentum projection $\hslash L$ in units of $\hslash$. Excitation paths are shown by arrows. Central: squared overlap (fidelity) of $\mid \psi \left(t\right)⟩$ with the initial (solid lines) and final (dashed lines) states as a function of the duration of a square pulse. Right: fidelities as a function of time for an optimized Gaussian pulse of the form $e^{-{(t-t_0)}^2∕{\tau }^2}$. Time is measured in units of ${\tau }_0=\hslash ∕U\sim {10}^{-4}\mathrm{s}$. For $n=2$, the peak perturbation amplitude is $V_p=0.05(U∕2)$, $\omega -{\Omega }_p=2.0(U∕2)$, and a Gaussian pulse time of $\tau =24{\tau }_0$. For $n=3$, $\tau =102{\tau }_0$ and $\omega -{\Omega }_p=2.046(U∕2)$ and $2.055(U∕2)$ for the Gaussian and square cases, respectively. For $n=3$, nonlinear effects (coupling with near-resonant levels) shifted the optimal frequency away from the linear response expectation, $\omega -{\Omega }_p=2(U∕2)$.

Figure 2

(Color online) Using a rotating $m$-fold symmetric perturbation to drive $n=3$ particle clusters from $L=2$ to the $\nu =1∕2$ Laughlin state. Left: path on the energy level diagram. Center: second-order process coming from a deformation with $m=2$. Right: direct transition produced with $m=4$. Solid (dashed) lines are fidelities with the initial (Laughlin) state. In both cases the peak deformation is $V_p=0.05(U∕2)$. Both use a Gaussian pulse. The frequencies and pulse times $\tau$ we used for $m=2,4$ were $\omega -{\Omega }_p=(3.00∕2)(U∕2)$, $3.035(U∕2)$ and $\tau ∕{\tau }_0=218,21$. Note how much more rapid the direct process is.

Figure 3

(Color online) Transferring atoms using multiple pulses. Left: paths from initial to Laughlin states for $n=3,4$. Right: solid line is the fidelity with the initial state, dotted line with the intermediate $(L,E)=\mathbf{(}2,3(U∕2)\mathbf{)}$ state, and the dashed line with the Laughlin state. All pulses are Gaussians. Despite using multiple pulses, this technique is faster than using a higher-order $m=2$ pulse. The frequencies $\left({\Omega }_p\right)$, shape $\left(m\right)$, and pulse times $\left(\tau \right)$ for the $N=3$ sequence were $\hslash (\omega -{\Omega }_p)∕(U∕2)=3.00,3.035$, $m=2,4$, and $\tau ∕{\tau }_0=16.95,19.2$. For both, $V_p=0.05(U∕2)$. For $N=4$, using two pulses with $m=2$ and $V_p=0.2(U∕2)$, we achieve $>98\%$ fidelity after a total of two-pulse sequences with $\hslash (\omega -{\Omega }_p)∕(U∕2)=3.130,1.0376$ and $\tau ∕{\tau }_0=82.5,87.0$.