Realization of a Quantum Integer-Spin Chain with Controllable Interactions

The physics of interacting integer-spin chains has been a topic of intense theoretical interest, particularly in the context of symmetry-protected topological phases. However, there has not been a controllable model system to study this physics experimentally. We demonstrate how spin-dependent forces on trapped ions can be used to engineer an effective system of interacting spin-1 particles. Our system evolves coherently under an applied spin-1 $XY$ Hamiltonian with tunable, long-range couplings, and all three quantum levels at each site participate in the dynamics. We observe the time evolution of the system and verify its coherence by entangling a pair of effective three-level particles (“qutrits”) with 86% fidelity. By adiabatically ramping a global field, we produce ground states of the $XY$ model, and we demonstrate an instance where the ground state cannot be created without breaking the same symmetries that protect the topological Haldane phase. This experimental platform enables future studies of symmetry-protected order in spin-1 systems and their use in quantum applications.

Popular Summary

Two isolated quantum states in an atom can be used experimentally to represent a quantum version of a bar magnet that has either its north or south pole pointing up. Coupling many such atoms together enables physicists to study how magnetism works at the quantum level. Controlling more than two states per atom is harder to achieve in the lab but would make it possible to experiment with more complicated forms of quantum magnetism that are exponentially harder to study on a computer ($3^N$ versus $2^N$). Here, we control interactions among ions with three quantum states apiece (“qutrits”): a north or south pole-up magnet or no magnet at all.

We employ trapped ${}^{171}{\mathrm{Yb}}^{+}$ ions; the energy gap of such a spin-1 system may be useful for quantum memories in the future. The quantum-mechanical interactions allow magnets to hop from site to site and allow pairs of magnets with different orientations to appear and disappear together. After preparing two empty sites, we optically measure the spin state and observe signatures of magnet pairs appearing and disappearing; we also show that the pairs exhibit quantum entanglement persisting for milliseconds. By manipulating a special magnetic field, we also prepare the quantum ground state (the configuration of magnets and empty sites that have the lowest possible interaction energy).

This new method for creating spin-1 quantum magnets opens the door to studying the fundamental physical properties of magnets and learning to exploit them for quantum technologies. Our method can be used for other types of interactions and might even be a stepping stone toward creating exotic phases of matter that retain their quantum properties when they interact with the classical world.

Figure 1

(a) Level diagram for ${}^{171}{\mathrm{Yb}}^{+}$, highlighting relevant states. (b) Sketch of experimental geometry, showing the directions of the laser wave vectors and the real magnetic field relative to the ion chain. Both beams are linearly polarized, one along the $\overrightarrow{B}$ field (providing $\pi$ light) and one orthogonal to the $\overrightarrow{B}$ field (providing an equal superposition of ${\sigma }^{+}$ and ${\sigma }^{-}$ light). Multiple beat notes are applied by imprinting multiple frequencies onto one beam (in this case, the $\pi$-polarized beam). (c) Detailed level diagram of the ${}^2{S}_{1/2}$ ground state, showing Raman beat notes in relation to Zeeman splittings and motional sidebands for the center-of-mass mode. Level splittings are not to scale.

Figure 2

Dynamics of two spin-1 particles evolving under the $XY$ Hamiltonian in Eq. (2). (a) We measure the populations for each ion to be in the state $|0⟩$. The probability of both ions to be in $|0⟩$ (black diamonds), only the left ion in $|0⟩$ (blue triangles), only the right ion in $|0⟩$ (green squares), and neither ion in $|0⟩$ (red circles) are plotted together. Similar plots for the $|-⟩$ state (b) and $|+⟩$ state (c) are also shown, with a different symbol order emphasizing that the red circles represent the same state in all three graphs. The dynamics resemble Rabi flopping between the state $|00⟩$ and the symmetric superposition $(|+-⟩+|-+⟩)/\sqrt{2}$. Solid lines represent theoretical predictions, with the only free parameters being the magnitude of the $S_z$ and $(S_z{)}^2$ gradients discussed in the text. In (c), the interaction $J_{1,2}$ drifts at long times compared to that estimated from Eq. (3), consistent with an observed drift in radial trap frequencies during the data collection. Error bars (1s.d.) show the statistical uncertainty based on 500 repetitions of the experiment.

Figure 3

(a) Illustration of the rotations performed before measuring the parity, where rotations by $\theta$ and $\phi$ are defined in Eq. (5). Also shown are the ideal initial state and the states produced after each step. (b) The parity of the final state oscillates as a function of the final pulse phase $\phi$. Fitting the function $C-A\cos 2\phi +B\sin 2\phi$ (red dashed line) to the data results in an amplitude $A=0.86>0.5$, demonstrating entanglement. (c) Amplitude of the parity oscillation after various durations of the spin-spin interaction, showing the peak amplitude for each time $\sim (2n+1)/(2\sqrt{2}{J}_{1,2})$. The data in (b) correspond to the highest-contrast point in (c). The dashed line is a guide to the eye, suggesting the expected behavior of sinusoidal oscillations between a product state and an entangled state, along with decay due to decoherence at longer times. The chosen durations are not evenly spaced past 0.6 ms: for each point, the duration is chosen such that the population in $|00⟩$ is at a local minimum, and drifts in the radial trap frequencies lead to small changes in $J_{1,2}$.

Figure 4

Measurements of the prepared two-spin states (narrow blue bars) after ramping an $(S_z{)}^2$ field, compared to the values expected for the calculated ground state (gray bars). As in Fig. 2 show the measured populations when the dark state is set to be $|0⟩$, $|-⟩$, or $|+⟩$, respectively.

Figure 5

Measurements of the prepared four-spin states (narrow blue bars) after ramping an $(S_z{)}^2$ field, compared to the values expected for the calculated ground state (gray bars). Again, (a)–(c) show the measured populations when the dark state is set to be $|0⟩$, $|-⟩$, or $|+⟩$, respectively. Here, the interaction range is given by $J_{i,j}\sim 1/|i-j{|}^{0.3}$.

Figure 6

Following an adiabatic ramp, the parity of the final state is measured as a function of the final rotation phase $\phi$ (see text for rotation protocol). Dashed and dot-dashed lines represent the theoretically expected values for the ground state, $|00⟩/\sqrt{2}-(|-+⟩+|+-⟩)/2$, and highest excited state, $|00⟩/\sqrt{2}+(|-+⟩+|+-⟩)/2$, respectively. The phase of the oscillation reveals that the relative phases in the prepared state are consistent with the expected ground state.

Figure 7

Measurements when the $|0⟩$ state is set dark (a), the $|+⟩$ state is dark (b), and the $|-⟩$ state is dark (c), of the prepared three-spin state after adiabatically ramping a global $(S_z^i{)}^2$ field (narrow blue bars). The data agree closely with the calculated populations in the first excited state (gray bars), while showing little overlap with the expected populations in the ground state (wide, hatched red bars).