Research ArticleCONDENSED MATTER PHYSICS

Enhanced quantum coherence in exchange coupled spins via singlet-triplet transitions

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Science Advances  09 Nov 2018:
Vol. 4, no. 11, eaau4159
DOI: 10.1126/sciadv.aau4159
  • Fig. 1 Spin resonance for two coupled spin-1/2 Ti atoms.

    (A) Schematic of the ESR-STM setup with the topographic image of a pair of Ti atoms on 2 ML MgO, where the Ti atoms are separated by r = 0.92 nm. The two species appear with different apparent heights in the STM image: ~1 Å for Ti at the O binding site of MgO (TiO) and ~1.8 Å for Ti at a bridge site (TiB) (VDC = 40 mV, I = 10 pA, T = 1.2 K). The external magnetic field is applied almost parallel to the surface. (B and C) ESR spectrum measured on TiO in a TiO-TiB dimer with (B) r = 0.92 nm and (C) r = 0.72 nm [VDC = 40 mV, T = 1.2 K, Bext = 0.9 T; (B) I = 10 pA, VRF = 30 mV; (C) I = 20 pA, VRF = 15 mV]. Insets: STM images of the TiO-TiB dimer used to measure each ESR spectrum. The grid intersections indicate the positions of oxygen atoms of the MgO lattice. The mark “x” shows the position of the tip in the ESR measurement. (D and E) Schematic energy level diagrams for two coupled spin-1/2 atoms. In (D), the Zeeman energy is larger than the interaction energy J between two atoms, leading to the triplet state as the ground state. In (E), the singlet state becomes the ground state when J is larger than the Zeeman energy. The resonance peaks in (B) and (C) are marked by the same colors as transition labels in (D) and (E), respectively.

  • Fig. 2 Singlet-triplet energy detuning of TiO-TiB dimers with different interaction energies.

    (A) Magnetic interaction energy determined from ESR measurements for TiO-TiB dimers with different atomic separations. The red line shows the exponential fit, indicative of Heisenberg exchange interaction. The slight deviation of the TiO-TiB interaction energy from the exponential fit is due to the contribution from the dipole-dipole interaction at larger distances. (B) The ESR frequency shift of the S-T0 transition (Embedded Image) for dimers with different J as a function of the magnitude of the tip field (Btip). Btip is calibrated for each tip that we used (section S4). For the dimers with J = 0.5, 0.8, and 3 GHz, the resonance frequencies are obtained by Embedded Image; for the dimers with J = 9 and 30 GHz, Embedded Image is directly obtained from ESR spectra. Strengthening the exchange interaction between Ti atoms protects the |S⟩ and |T0⟩ states from detuning by Btip, reducing Embedded Image. (C) First-order tip field dependence of Embedded Image for the dimers in (B). The clock transitions appear at Btip = 38 ± 12 mT, where Embedded Image.

  • Fig. 3 Spin coherence of ESR transitions and their sensitivity to external and local magnetic fields.

    (A) ESR peak width as a function of VRF for the S-T0 and T0-T transitions measured on TiO in a strongly coupled dimer (r = 0.72 nm, J ≈ 30 GHz), and the |0⟩ to |1⟩ transition of an individual TiO atom (VDC = 40 mV, I = 10 pA, Btip = 110 mT, Bext = 0.9 T, T = 1.2 K). Solid lines are fits to Embedded Image, derived from the Bloch equation model, where the spin coherence time Embedded Image is determined by the intercept at the y axis and A is a constant. (B) ESR frequencies as a function of the external magnetic field Bext. For the S-T0 transition, the frequency Embedded Image remains almost constant, characteristic of a clock transition. Inset: Energy diagram for the four eigenstates at different Bext (VDC = 40 mV, I = 10 pA, Btip = 110 mT, T = 1.2 K). (C) ESR frequencies as a function of the tip magnetic field Btip. Btip is set by the junction impedance (VDC/I) and calibrated from the fit (red curves; see also section S4). For the S-T0 transition, the frequency Embedded Image remains almost constant and measurably increases when Btip is larger than 150 mT, which reflects the change of eigenstates from the ideal singlet and triplet states. Inset: Energy diagram at different Btip (VDC = 40 mV, I = 10 pA, Bext = 0.9 T, T = 1.2 K).

  • Fig. 4 Homodyne detection and enhanced spin coherence of the S-T0 transition.

    (A) The tip field effects on ESR line shape of the S-T0 transition. The ESR spectra are normalized and vertically offset. (B) DC bias dependence of the ESR signals for the T0-T transition (top) and S-T0 transition (bottom). For the S-T0 transition, homodyne detection allows VDC to be decreased without losing signal intensity (Btip = 110 mT, VRF = 20 mV, T = 1.2 K). (C) Spin coherence time, Embedded Image, as a function of VDC. Red curves are reciprocal fits. At fixed junction impedance (VDC = 40 mV, I = 10 pA), Embedded Image increases with lowering VDC because of the reduction of tunneling electrons per unit time. For the S-T0 transition, setting VDC to zero provides further improvement in the spin coherence time by reducing the DC tunneling current. Labels #1 and #2 indicate different dimers (measured with different tips) with the same separation (r = 0.72 nm) to confirm the reproducibility of Embedded Image.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/11/eaau4159/DC1

    Section S1. Binding site of Ti atoms on 2 ML MgO/Ag(001)

    Section S2. ESR of individual Ti atoms

    Section S3. Magnetic interaction of Ti dimers

    Section S4. Spin Hamiltonian of two coupled Ti spins

    Section S5. Detection mechanism of ESR

    Section S6. Driving mechanism of ESR

    Section S7. Determination of spin coherence time

    Fig. S1. Tunneling current as a function of the tip-atom distance for TiB (blue) and TiO (red) on 2 ML MgO/Ag(001) at VDC = 10 mV.

    Fig. S2. ESR of individual Ti atoms.

    Fig. S3. Characterization of magnetic interaction between Ti atoms.

    Fig. S4. Tip field effects on eigenenergies and eigenstates of TiO-TiB dimers.

    Fig. S5. Resonance frequencies measured on each atom of a weakly coupled dimer (r = 0.92 nm, J ≈ 0.8 GHz) at different tip fields.

    Fig. S6. ESR frequencies measured on each atom of strongly coupled dimers (r = 0.72 nm, J ≈ 30 GHz) at different tip fields.

    Fig. S7. ESR signals for the S-T0 transition with different DC biases.

    Fig. S8. Determination of spin coherence time at different DC biases and temperatures.

    Fig. S9. Spin coherence time with different DC biases (or different tunneling currents) for individual TiO and TiB atoms.

    Fig. S10. Comparison of ESR signals for the singlet-triplet transition measured on the TiO (black) and TiB (green) atoms in the TiO-TiB dimer.

  • Supplementary Materials

    This PDF file includes:

    • Section S1. Binding site of Ti atoms on 2 ML MgO/Ag(001)
    • Section S2. ESR of individual Ti atoms
    • Section S3. Magnetic interaction of Ti dimers
    • Section S4. Spin Hamiltonian of two coupled Ti spins
    • Section S5. Detection mechanism of ESR
    • Section S6. Driving mechanism of ESR
    • Section S7. Determination of spin coherence time
    • Fig. S1. Tunneling current as a function of the tip-atom distance for TiB (blue) and TiO (red) on 2 ML MgO/Ag(001) at VDC = 10 mV.
    • Fig. S2. ESR of individual Ti atoms.
    • Fig. S3. Characterization of magnetic interaction between Ti atoms.
    • Fig. S4. Tip field effects on eigenenergies and eigenstates of TiO-TiB dimers.
    • Fig. S5. Resonance frequencies measured on each atom of a weakly coupled dimer (r = 0.92 nm, J ≈ 0.8 GHz) at different tip fields.
    • Fig. S6. ESR frequencies measured on each atom of strongly coupled dimers (r = 0.72 nm, J ≈ 30 GHz) at different tip fields.
    • Fig. S7. ESR signals for the S-T0 transition with different DC biases.
    • Fig. S8. Determination of spin coherence time at different DC biases and temperatures.
    • Fig. S9. Spin coherence time with different DC biases (or different tunneling currents) for individual TiO and TiB atoms.
    • Fig. S10. Comparison of ESR signals for the singlet-triplet transition measured on the TiO (black) and TiB (green) atoms in the TiO-TiB dimer.

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