Unraveling spin entanglement using quantum gates with scanning tunneling microscopy-driven electron spin resonance

TL;DR

This work tackles entanglement generation in a solid-state, atom-scale platform by implementing universal quantum gates with ESR-STM. It employs two exchange-coupled Ti atoms on MgO/Ag(100) and uses the TimeESR simulator to design and analyze Hadamard and CNOT gate sequences, achieving Bell-state generation with high fidelity before decoherence. The study quantifies entanglement via fidelity and concurrence, showing robust Bell-state formation on sub-μs to μs timescales but inevitable decoherence from tunneling currents. The results establish ESR-STM as a viable route for atom-scale quantum circuits on surfaces, while outlining scalability and coherence challenges for larger qubit arrays.

Abstract

Quantum entanglement is a fundamental resource for quantum information processing, and its controlled generation and detection remain key challenges in scalable quantum architectures. Here, we numerically demonstrate the deterministic generation of entangled spin states in a solid-state platform by implementing quantum gates via electron spin resonance combined with scanning tunneling microscopy (ESR-STM). Using two titanium atoms on a MgO/Ag(100) substrate as a model, we construct a two-qubit system whose dynamics are coherently manipulated through tailored microwave pulse sequences. We generate Bell states by implementing a Hadamard gate followed by a controlled-NOT gate, and evaluate its fidelity and concurrence using the quantum-master equation-based code TimeESR. Our results demonstrate that ESR-STM can create entangled states with significant fidelity. This study paves the way for the realization of atom-based quantum circuits and highlights ESR-STM as a powerful tool for probing and engineering entangled states on surfaces.

Figures (6)

• Figure 1: One qubit and two qubit ESR-STM schemes. (a) Atomic scheme fo the ESR-STM setup; one Ti atom ($S=1/2$) on two monolayers of MgO grown on Ag(100). The STM tip is an atomically sharp electrode placed on the Ti atom (designated the transport site), driving the electronic current through it. (b) Scheme of the two-qubit ESR-STM setup consisting of two exchange-coupled Ti atoms (each $S=1/2$) on the same substrate as (a), with the STM tip placed on the transport site. (c) Entanglement gate scheme using a single-qubit Hadamard gate on the second site, followed by a two-qubit CNOT gate with the second site as the control qubit. The effect of each gate for an input $|0\rangle \otimes |0\rangle$ is shown below the circuit. The final state of the depicted circuit is the Bell state $|\Phi^+\rangle=\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$.
• Figure 2: Two-qubit system stemming from two spin-1/2 sites weakly interacting and slightly detuned such that Zeeman-product states, Eq. (\ref{['eqn:Zeeman_states']}), are an excellent approximation to the four-level system. The single-qubit transitions between them are designated by their respective rates $\gamma_i$.
• Figure 3: Pulse sequence used in the input of TimeESR to produce an entangled $|\Phi^+\rangle$ Bell state. Colors indicate the position of the pulse within the sequence, corresponding with the sequences shown in Fig. \ref{['fig4']} and Fig. \ref{['fig5']}.
• Figure 4: Spin evolution during the quantum circuit execution. The top color bar represents a schematic of the four pulse regions described in Fig. \ref{['fig3']}. (a) Expectation value of the spin operator aligned to the locally-applied magnetic field $\hat{S}_{x}$ for each site; see the inset for a description of the principal axes. Initially the frequency is tuned to drive the second site to a superposition state. During this time, no operation is performed on the transport site, but the electronic current causes decoherence and the value of the spin slightly drifts away from $\expval{S^{x}_1}=-0.50$. At 281 ns, the CNOT gate is applied and both expectation values go to zero. (b) Expectation value of the spin operator aligned to the electrode's spin polarization (Z-axis) $\hat{S}_{z}$ for each site. The expectation value of $\expval{S^{y}}$ (not shown) follows the same pattern as $\expval{S^{z}}$. The profile of $\expval{S^{z}}$ tracks with the result of each pulse operation. As shown in the inset, all calculations are done in the lab frame, leading to oscillations at the Larmor frequency of the in-plane spin expectation values.
• Figure 5: Population of the different states, Fig. \ref{['fig2']}, during the quantum circuit execution (a) and the computed electronic current (b). The top color bar represents a schematic of the four pulse regions described in Fig. \ref{['fig3']}. Both graphs show the fast evolution taking place before the pulses are turned off at around 300 ns and the free evolution of the two spins is allowed. The population of the states can be easily identified with the expectation value of each single spin in Fig. \ref{['fig4']}.