Creating fractional quantum Hall states with atomic clusters using light-assisted insertion of angular momentum

We describe a protocol to prepare clusters of ultracold bosonic atoms in strongly interacting states reminiscent of fractional quantum Hall states. Our scheme consists in injecting a controlled amount of angular momentum to an atomic gas using Raman transitions carrying orbital angular momentum. By injecting one unit of angular momentum per atom, one realizes a single-vortex state, which is well described by mean-field theory for large enough particle numbers. We also present schemes to realize fractional quantum Hall states, namely, the bosonic Laughlin and Moore-Read states. We investigate the requirements for adiabatic nucleation of such topological states, in particular comparing linear Landau-Zener ramps and arbitrary ramps obtained from optimized control methods. We also show that this protocol requires excellent control over the isotropic character of the trapping potential.

Figure 1

(a) Scheme of the experiment, based on a cluster of bosonic atoms, strongly confined along $z$, and subjected along $x$ and $y$ to an isotropic harmonic trap of frequency $\omega$. Laguerre-Gauss laser beams are sent on the atomic sample along $-z$ to drive Raman transitions transferring orbital angular momentum. Laser polarizations are chosen linear along $x$ and $y$, and a quantization magnetic field is applied along $x$. (b) Scheme of the involved energy levels $∥m;j,k〉$, indexed by the spin projection $m$ and the quanta $j,k$ of motion along $x$ and $y$, in the presence of an isotropic harmonic trap (“left-right” basis). Angular momentum is injected using Raman transitions, which couple $∥m;j,k〉$ to $∥m-1;j+1,k〉$ states via the absorption of one ${\mathrm{LG}}_0^1$ photon, followed by the emission of one ${\mathrm{LG}}_0^0$ photon. The trap isotropy ensures that orbital angular momentum is conserved, and $k\ne 0$ states remain unpopulated. The Raman transitions are resonant for a detuning ${\delta }_{\mathrm{mod}}$ equal to the difference between the trap frequency $\omega$ and the Zeeman detuning ${\delta }_Z$ (negative on the figure). On the example pictured here of a spin $F=1$, an adiabatic ramp of the detuning ${\delta }_{\mathrm{mod}}$ across resonance leads to a transfer of two units of angular momentum per atom.

Figure 2

(a), (b) Energy levels $E$ (black lines) as a function of the detuning $\delta$, for $N=2$ (a) and $N=8$ (b) particles of spin $F=1/2$, which corresponds to a transfer of one unit of angular momentum per atom. The Raman coupling amplitude is set to $\mathrm{\Omega }=\tilde{g}/\left(2\pi \right)$. The system is prepared at rest, spin polarized in $m=F$, corresponding to the many-body ground state for $\delta <0$. The adiabatic detuning ramp connects it to the one-vortex state $\left|{\psi }_{\mathrm{v}}\right.〉$, without any level crossing. The red dashed lines correspond to the lowest energy of stationary states of the mean-field nonlinear Schrödinger equation (4). For $N=8$ the mean-field energy almost coincides with the many-body ground state. (c), (d) Overlap $\mathcal{O}$ (c) and fidelity $\mathcal{F}$ (d) between the one-vortex state $\left|{\psi }_{\mathrm{v}}\right.〉$ and the state prepared for a ramp of finite speed $\dot{\delta }$, as a function of the dimensionless parameter ${\mathrm{\Omega }}^2/\dot{\delta }$ (blue solid lines). The overlap and fidelity are defined according to Eqs. (2) and (5), respectively. The calculation is performed for $N=8$ particles, and the result is compared with the Landau-Zener prediction expected for noninteracting systems (dotted lines). The calculated fidelity coincides with the one obtained from a mean-field description [dashed red line in (d)].

Figure 3

(a) Overlap between the Laughlin state and the state reached for a detuning ramp of speed $\dot{\delta }$, for particle numbers $N=2,3,4$ [(top) green, (middle) red, (bottom) blue lines, respectively], and a coupling $\mathrm{\Omega }=\tilde{g}/\left(2\pi \right)$. The vertical lines indicate the three ramp speeds considered in (b) and (c). (b) Atom density corresponding to the state reached for ramp inverse speeds ${\mathrm{\Omega }}^2/\dot{\delta }=1.5,3,5$ (dotted, dashed, solid blue lines, respectively) and $N=4$. The atom density of the $N=4$ Laughlin state is indistinguishable from the solid line. (c) Density probability $P_2\left(r\right)$ of the interparticle distance $r$ calculated for the three ramp speeds.

Figure 4

(a) Overlap between the Laughlin state and the state reached for an optimized ramp of duration $T$, either linear (crosses) or of arbitrary shape (dots), for atom numbers $N=2,3,4$ [(top) green, (middle) red, and (bottom) blue, respectively)]. The vertical lines indicate the quantum speed limits $T_{\mathrm{QSL}}$. Error bars (most of them smaller than the point size) are calculated for the ramps of arbitrary shape, as the error from the extrapolation to an infinite number of time discretization steps. (b) Quantum speed limit $T_{\mathrm{QSL}}$ deduced from (a) as a function of the atom number $N$.

Figure 5

(a) Overlap between the Moore-Read-like state $\left|{\psi }_{\mathrm{MR}}^{*}\right.〉$ and the state reached for a detuning ramp of speed $\dot{\delta }$, for particle numbers $N=4,6$ [(top) blue, (bottom) brown lines, respectively], and a coupling $\mathrm{\Omega }=\tilde{g}/\left(2\pi \right)$. The vertical lines indicate the two ramp speeds considered in (b). (b) Density probability $P_3\left(R\right)$ for the three-body hyperradius $R$, calculated for the Moore-Read state $\left|{\psi }_{\mathrm{MR}}\right.〉$, the actual ground state $\left|{\psi }_{\mathrm{MR}}^{*}\right.〉$, and the states reached after detuning ramps of inverse speed ${\mathrm{\Omega }}^2/\dot{\delta }=1,3.2$ (dash-dotted and dotted black lines, dashed and solid brown lines, respectively), in the case $N=6$.

Figure 6

(a) Overlap $\mathcal{O}$ between the Laughlin state $\left|{\psi }_{\mathrm{L}}\right.〉$ and the ground state of the full Hamiltonian for $N=3$ (red line), a Raman coupling $\mathrm{\Omega }=\tilde{g}/\left(2\pi \right)$, and ellipticities $\epsilon =0$ (dashed line) and $\epsilon =0.05\tilde{g}/\left(2\pi \right)$ (solid line), as a function of the detuning $\delta$. A maximum value ${\mathcal{O}}_{\max }$ can be reached when varying the detuning $\delta$. (b) Maximum overlap ${\mathcal{O}}_{\max }$ as a function of the ellipticity $\epsilon$.

Figure 7

(a) Optimum overlap ${\mathcal{O}}_n$ as a function of the inverse number of discretization steps $n^{-1}$, for $N=4$ and $T=5$. The optimum overlap $\mathcal{O}$ is obtained from the extrapolated value ${\text{lim}}_{n\to \infty }{\mathcal{O}}_n$, obtained from a linear fit of the data for $n\ge 50$. (b) Example of optimum detuning ramp ${\delta }_{\mathrm{opt}}\left(t\right)$ corresponding to $N=4$ and $T=5\times 2\pi /\tilde{g}$, with $n=200$ discretization steps. (c) Optimum overlaps ${\mathcal{O}}_{\mathrm{opt}}$ calculated for $N=2$, 3, and 4 [(top) green, (middle) red, and (bottom) blue, respectively]. The quantum speed limit times $T_{\mathrm{QSL}}$ are obtained from piecewise linear fits (solid lines).