Efficient driving of a spin-qubit using single-atom magnets

TL;DR

The paper demonstrates that a nearby single-atom magnet can efficiently drive a spin-qubit via electric-field–induced modulation of exchange coupling. By developing a three-orbital superexchange model, it shows that $J(\mathbf{E})$ can be tuned so that $J_1$ produces Rabi rates in the tens of MHz range, consistent with experiment. Density functional theory supports these findings, providing quantitative values for $J_0$ and the field slopes $J_1$, while showing that adatom displacement from electric fields is negligible. The work identifies key experimental knobs, such as the static exchange $J_0$ and the external field angle, and establishes exchange-modulation as a principal all-electrical mechanism for ESR-STM control of surface-spin qubits with potential for scalable quantum sensing and manipulation.

Abstract

The realization of electron-spin resonance at the single-atom level using scanning tunneling microscopy has opened new avenues for coherent quantum sensing and quantum state manipulation at the ultimate size limit. This allows to build many-body Hamiltonians and the study of their complex physical behavior. Recently, a novel qubit platform has emerged from this field, raising questions about the driving mechanism from single-atom magnets. In this work, we demonstrate how single-atom magnets can be used to drive a nearby single spin qubit efficiently. We show that the modulation of exchange coupling is the primary driving force, which successfully reproduces Rabi rates in the tens of MHz range, consistent with experimental data, while also addressing critical aspects related to the optimization of experimental parameters.

Figures (11)

• Figure 1: Spin-qubit system: (a) Front view of two atoms subject to an electric field in the STM junction. External magnetic field direction as indicated. The atom-pair is exchange coupled with coupling strength $J$. (b) Spin qubit, i.e. a spin$-1/2$ without magnetic stability $D_0>0$. (c) Single-atom magnet $S>1/2$ with an energy barrier $D_0<0$. (d) Substrate-mediated superexchange of an atomic pair. Atom $1$ and $2$ are connected via the substrate wave-function represented by a single effective orbital $3$. $\ell_{1(2)}$ is the vertical distance from orbital 1(2) to the effective orbital 3. (e) $t_{1(2)}$ represents the hopping amplitude from orbital 1(2) to effective orbital 3 and $\Delta$ the site potential difference.
• Figure 2: Achievable Rabi rates from $D$, $J$ and $DJ$ modulation models: (a), (b) Dependence of the Rabi rate by separate tuning of $D$ and $J$ on keeping the respective other parameters fixed. (c) Dependence of the Rabi rate on $J_1$ and $D_1$ while choosing $D_0$ and $J_0$ values compatible with the physical Fe-Ti system. Fixed parameters are as follows: (a) $J_0 = 5 \textrm{ GHz}$ and $J_1 = 0 \textrm{ GHz}$; (b) $D_0 = -5$ meV and $D_1 = 0 \textrm{ meV}$; and (c) $D_0 = -5$ meV and $J_0 = 5 \text{ GHz}$. Parameters related to the external magnetic field are: $B_\text{ext} = 0.9 \text{ T}$, $\theta_\text{ext}= 80^{\circ}$. We use isotropic $g$-tensor $g=2$.
• Figure 3: Spin dynamics of a Fe-Ti pair using $DJ$-modulation: (a) Illustration of the static net magnetic field composed of the external field $\bm{B_\textrm{ext}}$ and the field induced by the spin-up state of Fe, $\bm{B}_{0, \textrm{Fe} (\Uparrow)}$, acting on the Ti spin. $\bm{B}_{1, \textrm{Fe} (\Uparrow)}$ is the time-dependent driving field induced by Fe spin dynamics. Time-dependence of (b) the Ti spin in the rotating frame and (c) the Fe spin and their representations on a Bloch sphere. The arrows indicate the initial spin configurations of Ti and Fe. As we can see, the Fe spin does not undergo any measurable time dependence. The transition frequency is $\omega_{ij} /(2 \pi) \approx 26 \text{ GHz}$, and the Rabi rate is $\Omega / (2 \pi)= 9.41 \text{ MHz}$. Simulation parameters: $D_0 = -4.7$ meV, $D_1 = 0.41 \mu\text{eV}$, $J_0 = 6.8 \text{ GHz}$, $J_1=0.01 \text{ GHz}$. $B_\text{ext} = 0.9 \text{ T}$, $\theta_\text{ext}= 80^{\circ}$. We employ isotropic $g$-tensor $g=2$.
• Figure 4: Optimization of Rabi rates. (a) Dependence of Rabi rates on the static coupling strength $J_0$ and the external magnetic field angle $\theta_\text{ext}$. Here we use a fixed ratio of $J_1/J_0=1/680$. (b) Dependence of the Ti spin's angle $\theta_\textrm{Ti}$ on the external magnetic field angle $\theta_\text{ext}$. Each line corresponds to a fixed value of $J_0$, ranging from 1 GHz to 20 GHz in steps of 1 GHz. Blue dots indicate the maximum Rabi rates. The remaining parameters are $B_\mathrm{ext}=0.9$ T, isotropic $g$-tensor $g=2$, $D_0=-4.7$ meV and $D_1=0.41\ \mu$eV.
• Figure S1: Three-orbital superexchange model: (a) Energy scheme of orbitals. Orbitals 1 and 2 describe the spin centers (i.e. adsorbates), whilst 3 is the effective substrate orbital. (b) Real-space schematic of the orbitals where we treat the insulating layer as an effective orbital $3$. $\ell_{1(2)}$ denotes the distance between orbitals and $\ell \equiv \ell_1 + \ell_2$. We calculated the absolute values of the exchange coupling $J_0$ from our model. (c) $J_0$ in log-scale as function of the on-site repulsion $U$ and hopping values $t = t_1 = t_2$. (d) The same simulation but $t$ was converted in atom-dimer distance $d$ assuming an exponential dependence of $t=t_0 \exp{(-d/4\lambda)}$ with $\lambda$ being the decay length. In all calculations $\Delta_1 = \Delta_2 = \Delta = 0.1 \text{ eV}$ and only the region with $0.01 \text{ GHz}\leq J_0 \leq 100 \text{ GHz}$ is shown.