Engineering the Eigenstates of Coupled Spin-$1/2$ Atoms on a Surface

Quantum spin networks having engineered geometries and interactions are eagerly pursued for quantum simulation and access to emergent quantum phenomena such as spin liquids. Spin-$1/2$ centers are particularly desirable, because they readily manifest coherent quantum fluctuations. Here we introduce a controllable spin-$1/2$ architecture consisting of titanium atoms on a magnesium oxide surface. We tailor the spin interactions by atomic-precision positioning using a scanning tunneling microscope (STM) and subsequently perform electron spin resonance on individual atoms to drive transitions into and out of quantum eigenstates of the coupled-spin system. Interactions between the atoms are mapped over a range of distances extending from highly anisotropic dipole coupling to strong exchange coupling. The local magnetic field of the magnetic STM tip serves to precisely tune the superposition states of a pair of spins. The precise control of the spin-spin interactions and ability to probe the states of the coupled-spin network by addressing individual spins will enable the exploration of quantum many-body systems based on networks of spin-$1/2$ atoms on surfaces.

Figure 1

ESR of a single Ti atom on MgO. (a) Schematic of the measurement setup showing an STM image of a hydrogenated Ti atom on bilayer MgO on Ag (001) and a magnetic tip. The black arrows indicate the orientations of the magnetic moments. A radio-frequency voltage is applied to drive ESR of Ti. (b) Left panels: Ball model of hydrogenated Ti on MgO and calculated spin density. Right panel: Schematic of the orbital occupancy of the $3d^1$ configuration. (c) ESR spectrum of a single Ti atom. The peak is fitted to an asymmetric Lorentzian [Eq. (S1) 21] ($V=50\mathrm{mV}$, $I=10\mathrm{pA}$, $V_{\mathrm{r}\mathrm{f}}=18\mathrm{mV}$, and $T=1.2\mathrm{K}$).

Figure 2

Engineering magnetic couplings between two Ti atoms. (a) Schematic of the measurement of Ti-Ti couplings. (b) Top panels: Positions of assembled Ti dimers are labeled ($n$, $m$) giving the number of unit cells separating them in increments of the oxygen lattice (lattice constant, 2.88 Å). Gray circles represent oxygen atoms. Bottom panels: The corresponding ESR spectra ($V=50$, 50, and 40 mV, $I=1$, 1, and 7 pA, $V_{\mathrm{r}\mathrm{f}}=22$, 22, and 30 mV, and $T=1.2\mathrm{K}$). The measured ESR splitting is shown. (c) Positions of all measured Ti dimers on MgO. Each of the yellow Ti constitutes one position relative to the center (blue) Ti. Note that only eight dimers are needed to be assembled in order to measure the 16 unique relative positions shown in (c). (d) ESR splitting of 11 assembled dimers as a function of azimuthal angle $\theta$ in (c). Negative values correspond to FM coupling. The black curve is the fit to the model Hamiltonian $H$ [see also Fig. S7(a) 21], tracing along a square that contains the yellow Ti in (c).

Figure 3

Tuning the quantum eigenstates of two coupled spins. (a) Effective tip magnetic field as a function of the tip height. Zero tip height corresponds to the junction resistance of $1\mathrm{G}\mathrm{\Omega }$ ($V=40\mathrm{mV}$, $I=40\mathrm{pA}$, and $T=1.2\mathrm{K}$). Inset: ESR frequency of an isolated Ti atom as a function of the tip height. The asymptotic value (22.37 GHz) describing the absence of the STM tip is indicated by the red dashed line. The solid lines are exponential fits. (b) Schematic energy level diagram of the two Ti spins as a function of $B$ and $B_{\mathrm{tip}}$ for given $J$ and $D$. (c) ESR spectra on one of the Ti spins of the (0, 3) dimer at three different tip fields ($V=40\mathrm{mV}$, $I=10$, 7, and 4 pA, $V_{\mathrm{r}\mathrm{f}}=30–40\mathrm{mV}$, and $T=1.2\mathrm{K}$). Spectra are normalized with respect to peak III and vertically offset for clarity. (d) $f_{\text{I}\mathrm{I}\mathrm{I}}-f_{\mathrm{I}}$ as a function of $B_{\mathrm{tip}}$ (black points). The black curve is a fit to the Hamiltonian $H$ in the main text. Arrows show positions of the spectra in (c). The green curve is the calculated interaction ratio $\eta$. The inset shows a schematic of the avoided level crossing.