Quantum simulation of superexchange magnetism in linear ion crystals

We present a system for the simulation of Heisenberg models with spins $s=1/2$ and $s=1$ with a linear crystal of trapped ions. We show that the laser-ion interaction induces a Jaynes-Cummings-Hubbard interaction between the atomic $\mathsf{V}$-type level structure and the two phonon species. In the strong-coupling regime the collective atom and phonon excitations become localized at each lattice site and form an effective spin system with varying length. We show that the quantum-mechanical superexchange interaction caused by the second-order phonon hopping processes creates a Heisenberg-type coupling between the individual spins. Trapped ions allow control of the superexchange interactions by adjusting the trapping frequencies, the laser intensity, and the detuning.

Figure 1

The $\mathsf{V}$-type three-level system consists of the ground state $|g\rangle$ and two metastable excited states $|e_1\rangle$ and $|e_2\rangle$. Two laser beams with properly chosen frequencies and polarizations create JC couplings between the $\mathsf{V}$-type level structure and the two radial $x$ and $y$ phonon species.

Figure 2

(a) The on-site phonon frequencies $\delta {\omega }_{\beta ,j}$ and (b) Coulomb-mediated long-range hopping amplitudes $t_{11,j}^{\beta }$ in the radial $x$ and $y$ directions in units of ${\omega }_z$ for a linear ion crystal with $N=21$ ions as a function of the ion positions. The aspect ratios are ${\omega }_x/{\omega }_z=50$ and ${\omega }_y/{\omega }_z=100$.

Figure 3

(a) The energy splitting $E_{-,x}-E_{-,y}$ versus $\delta$. The spin-phonon couplings satisfy $g_y=\sqrt{2.5}{g}_x$. (b) The energy difference $U=E_{\mathrm{exc}}-2E_{-,y}$ versus the detuning $\delta$. Here $E_{\mathrm{exc}}$ is the energy for the state with two excitations in one site and none in another. The three curves are the energy differences with respect to the three low-energy states.

Figure 4

Superexchange interaction in a system of three ions couples the states $|\uparrow \downarrow \uparrow \rangle$, $|\uparrow \uparrow \downarrow \rangle$, and $|\downarrow \uparrow \uparrow \rangle$ according to the effective Hamiltonian (18). We compare the probability of finding the system in states $|\uparrow \downarrow \uparrow \rangle$ and $|\uparrow \uparrow \downarrow \rangle$ computed by the effective Hamiltonian (18) (red circles and blue triangles) and Hamiltonian (10) (solid lines). The population of state $|\downarrow \uparrow \uparrow \rangle$ is indistinguishable from that of $|\uparrow \uparrow \downarrow \rangle$ and it is not shown in the figure. We assume axial trap frequency ${\omega }_z/2\pi =120$ kHz and aspect ratios ${\omega }_y/{\omega }_x=1.8$ and ${\omega }_y/{\omega }_z=100$. The parameters are set to $t_{1,2}^x/2\pi =0.86$ kHz, $t_{1,2}^y/2\pi =0.48$ kHz, $t_{1,3}^x/2\pi =0.1$ kHz, $t_{1,3}^y/2\pi =0.06$ kHz, $g_x/2\pi =19$ kHz, $g_y/2\pi =20$ kHz, and $\delta /2\pi =-0.22$ kHz, which ensure that the system is in the strong-coupling regime. The phonon detuning is set to $\delta {\omega }_{\beta }={\omega }_{\beta ,1}$, such that we have $\delta {\omega }_{x,1}-\delta {\omega }_{x,2}=2\pi \times 8$ Hz and $\delta {\omega }_{y,1}-\delta {\omega }_{y,2}=2\pi \times 4$ Hz.

Figure 5

The anisotropy ${\lambda }_{1,2}=J_{1,2}^z/J_{1,2}^{xy}$ in a system of three ions. (a) The anisotropy as a function of $g_y$. The parameters are set to $g_x/2\pi =12$ kHz and $\delta /2\pi =-0.5$ kHz. (b) We fixed the coupling $g_y/2\pi =18$ kHz and varied the detuning $\delta$. The hopping amplitudes are set to $t_{1,2}^x/2\pi =0.5$ kHz and $t_{1,2}^y/2\pi =0.7$ kHz.

Figure 6

Coherent superexchange interaction in a system of two ions. (a) We plot the time evolution of states $|1,-1\rangle$ (blue circles), $|-1,1\rangle$ (red squares), and $|0,0\rangle$ (gray triangles) according to the effective Heisenberg-like Hamiltonian (20) compared with the JCHV Hamiltonian (10) (solid lines). The parameters are set to $g_x/2\pi =32$ kHz, $g_y/2\pi =34$ kHz, $\delta =0$, $t_{1,2}^x/2\pi =0.1$ kHz, and $t_{1,2}^y/2\pi =0.17$ kHz. (b) The same but the spin-phonon couplings are set to $g_x/2\pi =34$ kHz and $g_x=g_y$.