Chiral spin currents in a trapped-ion quantum simulator using Floquet engineering

The most typical ingredient of topologically protected quantum states is magnetic fluxes. In a system of spins, complex-valued interaction parameters give rise to a flux, if their phases do not add up to zero along a closed loop. Here we apply periodic driving, a powerful tool for quantum engineering, to a trapped-ion quantum simulator in order to generate such spin-spin interactions. We consider a simple driving scheme, consisting of a repeated series of locally quenched fields, and demonstrate the feasibility of this approach by studying the dynamics of a small system. An emblematic hallmark of the flux, accessible in experiments, is the appearance of chiral spin currents. Strikingly, we find that in parameter regimes where, in the absence of fluxes, phonon excitations dramatically reduce the fidelity of the spin model simulation, the spin dynamics remains widely unaffected by the phonons when fluxes are present. Our work provides a realistic experimental recipe to engineer the minimal building block of a topological quantum system with a currently existing ion trap apparatus.

Figure 1

Driving scheme. (a) Shaking protocol for a chain of three spins: The first sequence $\left(t<3\right)$ renders interactions between spin 1 (left ion) and 3 (right ion) complex valued. For $3<t<4$ and $4<t<5$ real-valued interactions between spins 1 and 2 (center ion), and 2 and 3 are acquired. The strength of nearest-neighbor interactions is suppressed by a factor 1/5, while second-neighbor interactions are suppressed by a factor 2/5. This allows one to obtain a situation as depicted in (b): All ions interact equally, encircling a magnetic flux $\mathrm{\Phi }$.

Figure 2

Fidelity of different descriptions. (a) For $\tau /\mathrm{\Delta }=1/4$, we evaluate the discrepancy between the ideal model as depicted in Fig. 1 and the exact Floquet Hamiltonian from the XX model as a function of the driving strength ${\mu }_0$. (b) We compare the dynamics in the Ising model $H_J$ (three solid lines for three ions, distinguished by color and by the numbers within the diamonds along the lines) and the XX model $H_{\text{XX}}$ (dashed-dotted lines). The diamonds (connected by dashed lines as a guide to the eye) mark the stroboscopic evolution at times $t=mT$. (c) For different values of the flux and at stroboscopic times $t=mT$, the deviations between Ising and Dicke dynamics is plotted, defined as $|\mathrm{\Delta }\langle {\sigma }_z\rangle {|}^2\equiv {\sum }_i{({\langle {\sigma }_z^i\rangle }_{\mathrm{Ising}}-{\langle {\sigma }_z^i\rangle }_{\mathrm{Dicke}})}^2$. Here, as in rest of the Rapid Communication, the system parameters are ${\delta }_{\mathrm{COM}}=2\pi \times 80\text{kHz}$ for the blue-detuning from the center-of-mass mode, mass $M=171$ amu, trap frequencies ${\omega }_{XY}=2\pi \times 5\text{MHz}$, and ${\omega }_Z=2\pi \times 900\text{kHz}$. The Rabi frequency is $\mathrm{\Omega }=2\pi \times 200\mathrm{kHz}$ and the recoil frequency is ${\omega }_{\mathrm{rec}}=2\pi \times 26\mathrm{kHz}$. This choice leads to a root mean square of the spin-spin interactions $J_{\mathrm{rms}}=2\pi \times 270\text{Hz}$. Field strengths are $B_0=J_{\mathrm{rms}}$ and ${\mu }_0=20J_{\mathrm{rms}}$.

Figure 3

Comparison between spin model dynamics and Dicke model dynamics. We plot the spin evolution in the Dicke model (solid lines) and the Ising model (dashed lines) for different parameters $\tau$, adjusting fluxes $\mathrm{\Phi }=2\pi \tau /\mathrm{\Delta }$: (a) $\tau =\mathrm{\Delta }/4,\mathrm{\Phi }=\frac{\pi }{2}$. (b) $\tau =3\mathrm{\Delta }/4,\mathrm{\Phi }=-\frac{\pi }{2}$. (c) $\tau =\mathrm{\Delta }/3,\mathrm{\Phi }=\frac{2\pi }{3}$. (d) $\tau =\mathrm{\Delta },\mathrm{\Phi }=0$. In each plot, the different ions are distinguished by different colors and boxed numbers next to the lines.

Figure 4

Chiral currents and energy spectrum. For a particle on a triangle with $|J_{ij}|=J$ and flux $\mathrm{\Phi }$, we plot (a) the chiral currents defined in Eq. (4), and (b) the eigenenergies, in units of $J$, as a function of the flux.

Figure 5

Phase measurement. (a) Spin evolution for shaking on the first and the third ion $\left({\mu }_0=10J_{\mathrm{rms}}\right)$, with the central ion detuned at any time, having perfect agreement between Dicke and Ising model dynamics. Different ions are distinguished by color and boxed number along the curves. (b) The complex phase is extracted from the wave-function dynamics in the Ising and the Dicke model, and from the correlation functions in the Dicke model. Theoretically expected values, for $\tau =\mathrm{\Delta }/3$, are marked by the black dash-dotted lines.